Blend tomatillos, onion, poblano pepper, jalapeno pepper, cilantro, garlic, mint, and salt in a blender until smooth. Transfer mixture to a saucepan and bring to a gentle boil. Reduce heat to medium-low and simmer until thickened, stirring often, about 10 minutes. Add chicken to tomatillo sauce and continue to simmer until chicken is heated through, about 5 minutes more.
Heat vegetable oil in a large skillet over medium heat. Fry tortilla strips in batches until crisp and lightly browned, 5 to 7 minutes. Drain on a paper towel-lined plate.
Layer about half the fried tortillas in a 2-quart baking dish; spread half the tomatillo sauce over tortillas. Mix pepperjack cheese and Monterey Jack cheese in a bowl; sprinkle half over tomatillo sauce. Repeat layers.
Bake in preheated oven until cheese is golden and bubbly, 30 to 35 minutes. Allow to cool 5 minutes before serving.