Heat 1 tablespoon vegetable oil in a large cast iron skillet over medium high heat. Add onion and bell peppers, and cook, stirring occasionally, until tender and lightly charred, about 4-5 minutes.
Stir in garlic, chili powder and cumin until fragrant, about 1 minute; season with salt and pepper, to taste. Set aside.
In a medium bowl, combine cheeses and cilantro.
Heat 1 teaspoon vegetable oil in the skillet over medium low heat. Working one at a time, add 1 tortilla to the skillet and sprinkle with 1/3 cup cheese mixture in an even layer.
Top with 1/3 cup chicken and some of the bell pepper mixture on half of the tortilla.
Cook until the cheeses have melted and the underside is golden brown, about 3-4 minutes; gently fold in half, pressing lightly. Keep warm in oven up to 20 minutes. Repeat with remaining vegetable oil and tortillas to make 6 quesadillas.