1 1/2 cups diced sweet onion
2 large poblano peppers, seeded and diced
2 tablespoons canola oil
3 garlic cloves, minced
2 (10 3/4-oz.) cans cream of chicken soup
1 (8-oz.) container sour cream
2 (4-oz.) cans chopped or diced green chiles
1 cup chicken broth
1 envelope taco seasoning mix
18 (6-inch) corn tortillas, quartered
6 cups shredded cooked chicken
2 cups shredded extra-sharp Cheddar cheese
2 cups shredded pepper Jack cheese
Preheat oven to 350°. Sauté first 2 ingredients in hot oil in a large skillet over medium-high heat 5 minutes or until tender. Add garlic, and sauté 1 minute. Stir in soup and next 4 ingredients; remove from heat.
Spread 1 cup soup mixture in a lightly greased 13- x 9-inch baking dish. Arrange 24 tortilla quarters, slightly overlapping, over soup mixture; top with 3 cups chicken, 3/4 cup each Cheddar and pepper Jack cheeses, and 1 1/2 cups soup mixture. Repeat layers once. Top with remaining 24 tortilla quarters, soup mixture, and cheeses.
Bake at 350° for 45 to 50 minutes or until bubbly and golden. Remove from oven to a wire rack, and let stand 15 minutes before serving.