For the stuffing: In a medium skillet, heat the olive oil over medium-high heat. Add the sausage. Cook and crumble with a wooden spoon until cooked through, about 5 minutes. Increase heat to high, add the white wine, and cook until absorbed, about 3 minutes. Scrape into a medium bowl, and crumble sausage with a fork until very fine and crumbly. Let cool completely.

To the sausage, add one egg, the grated cheese, and bread crumbs. Mix well.

For the olives: In a deep pot, heat 2 inches of vegetable oil to 365 degrees. Line a sheet pan with parchment.

Press the stuffing into the cavities in the olives, filling all the way to the top and packing the stuffing tightly. Put the flour, two eggs, and bread crumbs into three separate shallow bowls. Dredge the olives in flour, and then in the whisked eggs, letting the excess drip back into the bowl. Roll to coat them in the bread crumbs, and rest on the sheet pan while you bread the remaining olives.

When all of the olives are breaded, fry, in three batches, until the breading is crisp and golden, about 3 to 4 minutes per batch. Drain on paper towels, and season lightly with salt—the olives are already salty.