1. In a 5- to 6-quart nonstick pan over high heat, frequently stir onion, bell peppers, garlic, and cumin seeds in 1 teaspoon oil until the onion is limp, 5 to 7 minutes. Stir in corn, beans, and chopped cilantro; remove pan from heat.
2. Pour the chili sauce into a 12-inch rimmed pizza pan or 10-inch pie pan. Dip a tortilla in sauce to coat both sides lightly; lift tortilla out, letting excess sauce drip back into pan. Set tortilla in an oiled 10-inch cheesecake pan with removable rim at least 3 inches tall (see notes). Spread 1/7 (about 1 cup) of the vegetable filling level over tortilla. Sprinkle evenly with 1/3 cup jack cheese and 3 tablespoons cotija cheese. Repeat layers, making a total of seven; top with last tortilla. Reserve any leftover chili sauce and remaining jack and cotija cheeses. Cover pan with an oiled piece of foil, oiled side down. Set pan in a rimmed 10- by 15-inch baking pan.
3. Bake enchilada pie in a 350° regular or 325° convection oven for 30 minutes. Uncover and continue baking until hot (160°) in the center, 30 to 40 minutes longer.
4. Before serving, in a microwave-safe bowl, in a microwave oven on full power (100%), heat remaining chili sauce until hot, 20 to 30 seconds. Pit, peel, and slice avocado.
5. Run a knife between pie and pan rim to loosen. Remove rim, set pie on a platter, and drizzle with chili sauce. Arrange avocado slices in a ring on pie, sprinkle with remaining jack and cotija cheeses, and garnish with cilantro sprigs. Cut into wedges to serve.
For a nonvegetarian pie, substitute 2 1/4 cups shredded cooked chicken for the black beans. If cheesecake pan rim is less than 3 inches tall, fold a 36- by 12-inch strip of foil in half twice lengthwise and oil one side. Line rim with strip, oiled side in and edge extending above rim. Pie can be prepared through step 2 up to 1 day ahead. Cover and chill pie, remaining chili sauce, and cheese separately. Bake, covered, for 1 hour; uncover and bake until hot in the center, 40 to 50 minutes longer.