Put the tomatoes, chili powder, garlic, 1/2 teaspoon salt and 2/3 cup water in a small saucepan and bring to a simmer. Cook 10 minutes, breaking up the tomatoes with a spoon. Add the chicken and cook, flipping once, until cooked through, 8 to 9 minutes per side. Transfer the chicken to a plate, shred it, then return it to the sauce. Add salt and black pepper to taste.
Meanwhile, heat the oil in a large skillet over medium-high heat. Add the bell peppers and cook until tender and slightly charred, 8 to 10 minutes. Season with salt and black pepper.
Lay a tortilla on a work surface and top evenly with 1/2 cup of the Cheddar. Add one-quarter of the shredded chicken and sauce and one-quarter of the bell peppers. Fold the tortilla in half to enclose the fillings. Repeat with the remaining tortillas and ingredients.
Melt 1/2 tablespoon of the butter in a large nonstick or cast-iron skillet over medium heat. Cook a quesadilla, flipping once, until the cheese is melted and it is golden brown on both sides, about 3 minutes per side. Transfer to a cutting board and repeat with the remaining butter and quesadillas.
Cut into triangles and serve topped with sour cream, chopped scallions, guacamole and salsa.
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