Preheat the oven to 350 degrees F (175 degrees C). Spray a skillet with cooking spray and heat over medium heat. Cook mushroom patties in the pan until halfway cooked through, 7 to 9 minutes. Remove from heat and let cool, about 5 minutes. Chop finely on a cutting board. Line the bottom of a rectangular or square 2 1/2-quart baking dish with patty pieces. Pour chili on top and spread evenly. Layer potato nuggets over chili. Mix mushroom soup, asparagus soup, evaporated milk, onion, and pepper together in a large bowl. Pour mixture over potato nuggets evenly. Bake in the preheated oven until bubbly and heated through, 45 minutes to 1 hour, removing 15 minutes before baking completes to sprinkle with Cheddar cheese.