1. Have ready the flour and beaten eggs, each in a shallow bowl. Heat 1/4 cup of olive oil in a large skillet.
2. One at a time, dredge the eggplant slices in the flour and then dip them into the beaten eggs, allowing the excess to drip off.
3. Fry 2 coated eggplant slices at a time in the olive oil over moderate heat until golden brown, about 2 minutes per side. Transfer the slices to paper towels to drain. Add more olive oil to the skillet as needed and adjust the heat if the eggplant browns too fast.
4. Preheat the oven to 400°. Lay the eggplant slices out on a work surface and lightly season them on both sides with salt and pepper. Cover the eggplant slices with the prosciutto; trim the prosciutto as necessary and use the trimmings to cover any uncovered eggplant slices. Cut each eggplant slice in half crosswise. Set 1 strip of mozzarella at 1 end of each eggplant slice, roll up neatly and secure with a toothpick.
5. Transfer the eggplant rolls to a baking sheet and bake for about 10 minutes, or until the cheese is melted and the eggplant is tender and piping hot. Remove the toothpicks and cut the rolls crosswise into 1-inch lengths. Serve the roll-ups hot or warm.

Makes about 2 dozen.

Make ahead: the roll-ups can be prepared through Step 2 and refrigerated overnight. Bring to room temperature before baking.