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Heat a large skillet over medium-high heat; add 2 teaspoons of the vegetable oil. Add the chicken, sprinkle with salt and pepper and cook until lightly browned and cooked through, about 5 minutes. Transfer the chicken to a plate and, to the same skillet, add the peppers and onions. Season with salt and pepper and cook, stirring occasionally, until the onions begin to caramelize and the peppers have softened, 10 to 12 minutes. Transfer to a plate and set aside in a warm place.
Heat the remaining teaspoon of vegetable oil in a medium skillet over medium heat. Add 1 of the tortillas, top with 1/2 cup of the cheese and cook until the cheese is partially melted, 2 to 3 minutes. Top with one-quarter of the chicken, one-quarter of the peppers and onions and 2 tablespoons of the sweet Thai chili sauce. Fold the tortilla in half, turn the heat up to medium high and cook until golden brown and toasted, about 2 minutes per side. Repeat with the remaining tortillas. Cut the quesadillas in half and serve with chips and salsa on the side.