Instructions:

1. Place each chicken breast on large sheet of plastic wrap, cover with second sheet, and pound with meat pounder or rolling pin until ¼-inch thick throughout. Each pounded breast should measure roughly 6 inches wide and 8½ inches long. Cover and refrigerate while preparing Stuffing.
   
   
2. Place breasts skinned side down on work surface; season with salt and pepper. Fill, roll, and wrap each breast. Refrigerate until filling is firm, at least 45 minutes.
   
   
3. Adjust oven rack to lower-middle position; heat oven to 400 degrees. Spread flour in pie plate or shallow baking dish. Beat eggs with 1 tablespoon vegetable oil and 1 tablespoon water in second pie plate or shallow baking dish. Spread bread crumbs in third pie plate or shallow baking dish. Unwrap chicken breasts and roll in flour; shake off excess. Using tongs, roll breasts in egg mixture; let excess drip off. Transfer to bread crumbs; shake pan to roll breasts in crumbs, then press with fingers to help crumbs adhere. Place breaded chicken breasts on large wire rack set over jelly-roll pan.
   
   
4. Heat remaining ¾ cup oil in medium skillet over medium-high heat until shimmering, but not smoking, about 4 minutes; add chicken, seam side down, and cook until medium golden brown, about 2 minutes. Turn each roll and cook until medium golden brown on all sides, 2 to 3 minutes longer. Return chicken rolls, seam side down, to wire rack set over jelly-roll pan; bake until deep golden brown about 15 minutes. Let stand 5 minutes before slicing each roll crosswise diagonally into 5 medallions; arrange on individual dinner plates and serve.