Mix the tomato sauce, tomato paste, mushrooms, mozzarella cheese, and red pepper flakes together in a bowl.
Heat the vegetable oil in a skillet over medium heat. Stir in the green and red bell peppers, garlic, and onion; cook and stir until the onion has softened and turned translucent, about 5 minutes.
Drain and reserve the liquid from the vegetables; set aside.
Spread about 1/3 of the tomato sauce mixture into the bottom of a casserole dish.
Layer 1/2 of the vegetables on top, then repeat the layers, topping with the remaining 1/3 of sauce.
Drizzle the reserved liquid from the vegetables on top.
Bake in the preheated oven until the cheese is melted and the sauce is bubbly, about 45 minutes.