In a large skillet, heat butter with oil over medium-high heat. Cook mushrooms and onions, stirring occasionally, for about 8 minutes or until liquid is evaporated. Add apple and carrot, cook for 1 minute. Remove from heat. Add bread crumbs, parsley, thyme, sage, salt, pepper. Combine well. Set aside. Cut each tenderloin lengthwise halfway through and open like a book. Place between plastic wrap or waxed paper. Using a meat pounder or rolling pin, pound to generous 1/4-inch thickness. Cut about 7 pieces of kitchen string into 15-inch lengths. Arrange crosswise about 11/2 inches apart in lightly greased roasting pan. Place 1 piece of meat in strings. With hands, press half of the stuffing over meat to cover surface. Top with second piece of meat, placing wide end over thin end of first piece. Tuck thin ends under and press firmly in place. Top with remaining stuffing and meat. Tie strings around roast, trimming any excess. Spread mustard all over roast. Arrange apple slices in lengthwise row on top, sprinkle with salt and pepper. Arrange thyme sprigs over top. Place in preheated 400 degree oven, reduce heat heat to 350 degree. Roast for 1 1/4 to 1 3/4 hours or until meat thermometer registers 160 degrees. Transfer to cutting board and cover with foil. Let stand for 10 minutes. Slice roast and arrange on platter, surround with Pan-Seared Onion and Apples.
Cut uncored apples crosswise into scant 1/2-inch thick slices. Discard ends. Repeat with onion. In large skillet, heat 1/2 of the oil over medium heat. Sear apples and onion for 1 to 2 minutes per side or until browned, adding more oil if necessary.