Toast the bread crumbs in 2 tablespoons of the butter until golden brown. Let cool.

Making the squash and cranberry filling:

Put the squash, sugar, salt, lemon juice and water in a large saucepan, cover, and set over medium-high heat. When boiling, lower the heat to keep at an active, bubbling simmer. Cook the squash covered for about 20 minutes, or until the chunks are al dente: soft enough to pierce with a fork but still with a fairly firm shape. Lift them out of the liquid with a slotted spoon or spider, drain, and set in a bowl to cool. Boil the cooking liquid uncovered until it has reduced to 1/4 cup or so of syrup; scrape it into a cup or bowl to stop the cooking.
Meanwhile, rinse the cranberries, put them into a small saucepan and stir in the sugar and orange juice. (If using frozen berries, don’t thaw: rinse and start cooking while they’re still frozen.) Cover the pan and bring the liquid to a boil over medium heat, stirring frequently to make sure the sugar dissolves. Uncover and cook at a gentle simmer for about 25 minutes, until the sauce has thickened and the cranberries are thoroughly cooked (but still have their berry shape).
Let the berries and squash cool to room temperature, then fold them together gently, in a large mixing bowl, keeping the chunks intact. Fold in the apricot jam and the soft butter and finally the squash syrup—or as much syrup as needed—without making the filling too loose. It should be dense enough to stay in a mound when you heap it up, with no liquid seeping out.

Rolling out the dough and arranging it on the cloth:

Before you form the strudel, arrange a rack in the center of the oven and preheat to 375 degrees.
On a large, lightly floured work surface, roll and stretch the strudel dough into a rectangle that measures approximately 2 feet wide (left to right in front of you) and 18 inches high (from the nearest edge to the one farthest from you). Follow the strudel stretching procedures from the Homemade Strudel Dough recipe. Start by rolling the dough into a rectangular shape then gradually stretch it with your hands until it reaches the size you want.
Now you want to cover the work surface with the cloth that will help you roll the strudel. The best way is to have someone else lay out the cloth while you hold the dough up in the air (or lay out the cloth on another flat surface then transfer it). Arrange the cloth so that its front edge is aligned with the near edge of your counter (it can drape over the edge a few inches too). The cloth should extend at least 4 feet left to right and 3 feet toward the far side of the counter. When the cloth is flat and smooth, flour it lightly then lay down the sheet of dough with its long side left to right in front of you, parallel with the cloth. There should be 6-inches to a foot of uncovered cloth all around the dough.

Rolling up the strudel:

Brush the parchment liner of the baking pan with about 2 tablespoons of the soft butter; keep the sheet handy.
Brush the surface of the dough—gently—with soft butter all over, using about 2 tablespoons in all. Stir and toss together the toasted bread crumbs, the cinnamon, and about 2 teaspoons of the sugar. Sprinkle this evenly all over the buttered dough (it keeps the layers of dough separated when the strudels rolled up).

Now you’re going to create a long row of the squash-cranberry filling on the dough. The row will be about 18-inches long, running left to right in front of you, about 4-inches in from the edge of the dough. The row will start about 3-inches in from the left edge of the dough and end about 3-inches from the right edge of the dough. And it will be 3- to 4-inches high and 4 or 5-inches wide—leaving lots of dough to roll around it.
Got that? When you have a good idea of where the filling is going, scoop the filling out in cups all along the filling line onto the dough and use your hands to push it evenly into its proper place and shape, resembling a long, plump sausage of squash chunks in cranberry jelly. Wash and dry your hands. Now the rolling begins. Grasp the edge of the cloth nearest you, at the points close to the left and right corners of the dough. It is a good idea to have someone help you at least the first time, so each of you can take a side. Lift the cloth, bringing the 4-inch margin of uncovered dough up and against the row of filling. Roll the cloth into your fists, so your hands are close to the dough and in better control; with your knuckles tight into a fist facing the filling, lift and push your hands with cloth in it, forward, so the whole row of filling rolls over at the same time—and gets wrapped snugly in dough.

Keep gathering the cloth in your hands, holding it close to the strudel. Pull the cloth up gently as the strudel rolls over several more times, wrapping more dough around the filling, until you’ve created a long neat log. Get ready to roll the strudel right onto the baking sheet. First, make sure the seam of the dough is on top, so it will be on the bottom when you roll it onto the sheet: use the cloth to roll it seam-up. Next, place the wide edge of the baking sheet against the strudel and lift the far side of the pan so it looks like you’re going to roll the log up a steep ramp—get someone to help you hold the sheet.

Using the cloth give the strudel a final forceful roll so it goes over the rim and onto the buttered parchment liner, seam down, on the near side of the baking sheet. The empty ends of the strudel dough are probably flopping off the sides of the pan. Twist these flaps (several dough layers thick) so they are snug against the ends of the filling and cut them off, leaving just an inch or 2 of twisted dough sticking out of the strudel. Tuck them under the strudel log.

Form the strudel into a horseshoe shape, carefully curving each end in on the baking sheet. Brush the strudel dough with the remaining soft butter all over. Cut 1/2-inch slits for steam vents all over the top of the pastry: plunge the tip of a sharp, thin-bladed knife straight down into the strudel, making sure it pierces through the layers of dough. Finally, sprinkle the remaining teaspoon of sugar all over the top of the strudel.

Bake the strudel for approximately 45 minutes, rotating the pan front to back once or twice, for even baking. The strudel should be deeply browned with the juices of the filling bubbling out of the slits. If you want brush, any fruity juices in the baking pan on the strudel dough. You can also glaze the strudel with softened apricot jam, but wait until the pastry has cooled a bit. Glazing the hot strudel will cause the dough to lose all its crispness; glazing later on will soften it less. (I like crisp flakey dough, so I don’t glaze at all.)

Let the strudel cool on the pan briefly, and then use a wide metal spatula to separate it from the sticky, caramelized juices on the parchment sheet. Use two spatulas—and a helping hand—to lift up the strudel in one piece and transfer it to a cutting board or platter. Or just slice the strudel right in the baking pan for serving on individual dessert plates.

Serve slices of the strudel warm or at room temperature—with cream, whipped cream or ice cream if you like.