Preheat oven to 350 degrees. Spray an 8x12 or 9x13 baking dish with non-stick spray and set aside.
Melt butter over medium-low heat. Whisk in flour, garlic, and chopped chipotle peppers. Cook and stir for 2 - 3 minutes, stirring constantly. Gradually whisk in the milk a little at a time.
Increase heat to medium high, bring the mixture to a very low boil, stirring constantly. Remove from heat. Stir in the salsa verde and 1/2 cup of Monterrey Jack and 1/2 cup of the sharp cheddar. (Taste and season lightly with salt and pepper if needed. Keep in mind the cheddar is very salty.)
Let the cheese sauce mixture cool. Stir in the sour cream until smooth and creamy.
Spoon 1/2 cup of the cheese sauce into the bottom of the prepared baking dish. Reserve another 1 1/2 cups for later. Stir the cooked and chopped chicken into the remaining cheese sauce in the skillet.
Spoon a generous amount of cheesy chicken into each warmed tortilla. Carefully fold and place seam side down into the baking dish. Repeat until the pan is full.
Pour the reserved cheese sauce on top. Sprinkle generously with the remaining cheese.
Bake uncovered for 30 minutes. Broil for 2 - 3 minutes until the cheese is golden and bubbly.
Let stand 15 minutes before serving.