Step 1Prepare the chiles: Roast the chiles under the broiler or over a stove-top burner until the skin is charred on all sides. Peel the skin and seed the chiles, then cut lengthwise into long strips.
Step 2Heat the oven to 375 degrees.
Step 3In a large saute pan, heat 2 tablespoons oil over medium-high heat. Add the onions and cook, stirring frequently, until softened, about 5 minutes. Stir in the corn and chile strips, reduce the heat to low and continue to cook until the corn and chiles are warmed through. Remove from heat and set aside.
Step 4In a medium saucepan, heat the remaining tablespoon of oil over medium-high heat. Add the garlic and cook until aromatic, about 1 minute. Add the tomato sauce and vegetable broth and heat until hot. Remove from heat and set aside.
Step 5In a food processor, combine the sour cream with two-thirds of the cotija cheese and blend until creamy but lumpy (this can also be done by hand in a large bowl, using the back of a spoon).
Step 6Line the base of a 13-by-9-inch baking or casserole dish with 3 lasagna noodles. Add one-half of the corn-chile mixture, distributing evenly over the noodles. Dollop one-half of the sour cream-cotija mixture over the corn and chiles. Place 3 more noodles in the pan, and repeat with the remaining corn-chile mixture and sour cream-cotija mix.
Step 7Place the 3 remaining noodles in the pan. Sprinkle the shredded quesadilla cheese and remaining cotija cheese over the noodles. Using a spoon, drizzle the thinned tomato sauce evenly over the noodles in the dish.
Step 8Cover the dish with foil and bake the lasagna for 30 minutes. Remove the foil and continue to bake until the cheese is melted and golden, an additional 10 to 15 minutes.