How to Make It
Preheat oven to 400°F. Brush a 9-by-13-inch baking dish with 1 Tbsp. oil. Place flour, eggs and bread crumbs in separate shallow bowls. Season chicken with salt and pepper and dredge in flour, shaking off excess. Dip chicken in eggs, then place in bread crumbs, turning once and pressing so crumbs adhere. Place breaded cutlets on a parchment-lined tray in a single layer.
In a large skillet, heat 1 Tbsp. oil over medium heat. Brown chicken in batches, flipping once, 2 to 3 minutes per side, adding 1 Tbsp. oil to skillet between batches. Transfer browned cutlets to baking dish, overlapping pieces as necessary to fit. In same skillet, cook mushrooms, stirring, until golden brown, about 5 minutes.
Drizzle wine over cutlets, then top with mushrooms, peppers and cheese. Bake until chicken is no longer pink and cheese has melted, about 15 minutes. Place under a hot broiler to brown, if desired.