Heat soup with broccoli until broccoli is thawed. Cook carrots and onions in 1 Tbsp oil until lightly browned. Reduce heat to low. Add 1/4 cup water and cover. Simmer until cooked, about 15 minutes. Remove to a bowl. In 3 Tbsp oil, cook mushrooms until moisture is gone. Stir in carrot mixture. Mix cheeses and eggs. Set oven to 375F. Place 1 cup sauce in pan. Layer with 1/2 of noodles, 1/2 of cheese mixture, all of carrot mixture, 1/2 remaining sauce, remaining noodles, cheese and sauce. Cook for 45 minutes.