Combine the tomatoes, chili powder, garlic and 1/2 teaspoon salt with 2/3 cup water in a small saucepan. Bring to a simmer and 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 and return it to the sauce. Add salt and pepper to taste.
Heat 1 tablespoon of the butter in a 10-inch skillet over medium heat. Add the mushrooms, 1/2 teaspoon salt and a few grinds of black pepper and cook, stirring occasionally, until the mushrooms are dry, about 4 minutes. Transfer the mushrooms to a small bowl.
Lay 1 tortilla on a work surface and top evenly with 1/2 cup mozzarella. Add a quarter of the shredded chicken and sauce, and a quarter of the mushrooms. Fold the tortilla in half to enclose the filling. 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 one of the quesadillas, flipping once, until the cheese is melted and it is golden brown on both sides, about 3 minutes per side. Remove to a plate and repeat with the remaining quesadillas.
Sprinkle the quesadillas with chopped scallions and serve with guacamole, sour cream and salsa if using.