Instructions

For Filling: Adjust oven rack to lowest position; heat oven to 450 degrees. Toss mushrooms with olive oil, and season with salt and pepper; spread in single layer on jelly-roll pan and roast until mushroom liquid has almost completely evaporated and mushroom surfaces facing pan are browned, 12 to 15 minutes. Turn mushrooms with metal spatula; continue to roast until mushrooms are browned all over, 5 to 10 minutes longer. Set aside.
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, rosemary, and provolone; season with salt and pepper and set aside.
For Chicken: Following illustration 1 below, 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 8 ½ inches long.
Following illustrations 2 through 4 below, place breasts skinned side down on work surface; season with salt and pepper. Fill, roll, and wrap each breast, placing mushrooms on top of cheese mixture on each breast. Refrigerate until filling is firm, at least 45 minutes.
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.
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 and instant-read thermometer inserted into center of each roll registers 155 degrees, about 15 minutes. Let stand 5 minutes before slicing each roll crosswise diagonally into 5 medallions; arrange on individual dinner plates and serve.