In a large skillet over medium-high heat, heat olive oil. Add peppers and onion and season with salt and pepper. Cook until soft, 5 minutes. Transfer to a plate.
Heat remaining tablespoon vegetable oil over medium-high heat. Season chicken with spices, salt, and pepper and cook, stirring occasionally, until golden and cooked through, 8 minutes. Transfer to a plate.
Add 1 flour tortilla to skillet and top half of the tortilla with a heavy sprinkling of both cheeses, cooked chicken mixture, pepper-onion mixture, a few slices of avocado, and green onions. Fold the other half of the tortilla over and cook, flipping once, until golden, 3 minutes per side. Repeat to make 4 quesadillas.
Slice into wedges and serve with sour cream.