FOR CUTLETS: Place each chicken breast on large sheet of plastic wrap, cover with second sheet, and pound with rubber mallet, meat pounder, or rolling pin until 1⁄4 inch thick throughout. Each pounded breast should measure roughly 6 inches wide and 8 1⁄4 inches long. Cover and refrigerate while preparing filling.
Place cutlets smooth-side down on work surface; season with salt and pepper. Fill, roll, and wrap each breast. Refrigerate until filling is firm, at least 1 hour.
Adjust oven rack to lower-middle position; heat oven to 450° degrees. Spread flour in pie plate. Beat eggs with 1 tablespoon oil in second pie plate. Spread bread crumbs in third pie plate. 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 cutlets on large wire rack set over baking sheet and allow coating to dry about 5 minutes.
Heat remaining 3⁄4 cup oil in heavy-bottomed 10-inch nonstick 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. Transfer chicken rolls, seam-side down, to wire rack set over rimmed baking sheet; bake until deep golden brown and instant-read thermometer inserted into center of a roll registers 155 degrees, about 15 minutes. Let stand 5 minutes before slicing each roll crosswise diagonally with serrated knife into 5 medallions; arrange on individual dinner plates and serve.
FOR FILLING: Heat butter in medium skillet over low heat until melted; add onion and sauté, stirring occasionally, until deep golden brown, 15 to 20 minutes. Stir in garlic and cook until fragrant, about 30 seconds longer; set aside.
In medium bowl using hand mixer, beat cream cheese on medium speed until light and fluffy, about 1 minute. Stir in onion mixture, thyme, gorgonzola, walnuts, dried figs, and sherry; season with salt and pepper and set aside.