Preheat the oven to 375 degrees.

For the filling, cut the squash into quarters lengthwise, and scrape out the seeds. Brush a 13-by-9-inch baking dish with olive oil. Arrange the squash pieces cut side up in the dish, brush lightly with olive oil, and season with salt and pepper. Bake until the squash feels soft and is easily pierced with a knife, about 1 hour. Allow to cool.

Remove the skin from the squash, and place the flesh in a food processor; process until no longer chunky, just a few quick pulses. Transfer the squash to a sieve lined with cheesecloth, and drain into a bowl in the refrigerator overnight.

Transfer the drained squash to a mixing bowl. Stir in the burrata, grated cheese, and egg yolks. Then stir in the breadcrumbs to make a filling that is tight but not dry. Add more breadcrumbs, a tablespoon at a time, if the filling is too loose. Refrigerate the filling while you make the dough.

For the dough, put the flour and cocoa in a food processor, and pulse several times. Beat 3 eggs, the egg yolk, olive oil, and 2 tablespoons water in a spouted measuring cup. With the food processor running, add the liquid, and let it run until the dough forms a ball on the blade, about 30 seconds. If it’s too loose, add a tablespoon or two of flour; if still crumbly, add a teaspoon or two of cold water.

Let the machine knead the dough for about 10 seconds. Turn the dough out on a very lightly floured surface, and knead by hand for another 30 seconds or so, until it’s smooth, soft, and stretchy. Wrap, and rest the dough for 30 minutes. Line two baking sheets with kitchen towels, and dust with flour.

Cut the dough into four pieces. Keeping the pieces you are not immediately working with covered, roll each piece through a pasta machine set to the next-to-last setting (or the last setting, if you are comfortable with your machine- you want the dough to be quite thin, since you will be folding it over), to get strips that are about as wide as the machine and about 16 inches in length.

Lay one strip out on a very lightly floured surface, and dollop scant tablespoons of filling at about 3-inch intervals down the center of the strip (you will get about six ravioli per strip). Press the tops of the mounds lightly to flatten. Beat the remaining egg and brush a thin strip along the top, bottom, and side edges of the dough strip and in between each mound of filling. Fold the dough over and seal.

Using a round pasta cutter, cut full half-rounds and fold in slightly. Arrange the finished ravioli on a lightly floured towel-lined baking sheet. Make more ravioli from the remaining pieces of dough.

When ready to cook, bring a large pot of salted water to a boil. For the butter-and-sage sauce, heat the butter in a large skillet over medium heat until melted and just foaming. Gently lay the sage leaves in the pan, and heat until they crisp up, about a minute. Ladle in 1 cup boiling pasta water; stir the sauce, and simmer for about 2 minutes, to reduce the liquid by half. Season the sauce with salt and pepper, and keep warm over very low heat while you cook the ravioli.

Add the ravioli to the boiling water, and let return to a simmer, stirring gently. Cook until the edges of the pasta are tender but still firm to the bite, about 3 minutes once the water returns to a simmer.

Transfer the cooked ravioli to the sauce with a spider or slotted spoon. Toss gently to coat. Remove the skillet from the heat, sprinkle with the 1 cup grated Grana Padano, and toss gently. Sprinkle the cookies over the top, and serve. If plating individually, gently scoop up the ravioli with a large spoon and place in large shallow bowls, covering the bottom and layering the ravioli, not piling them up, or use a flat plate.