Heat the oil in a large nonstick skillet over medium-high heat. Add the chorizo and cook, stirring, until aromatic and browned, about 4 minutes. Reduce the heat to medium and add the bell pepper. Cook, stirring, until slightly softened, about 3 minutes. Add the spinach and toss until wilted. Cool slightly.
Lay out 4 of the tortillas and sprinkle with half of the cheese (about 1/4 cup per tortilla). Divide the chorizo mixture among them, then sprinkle with the remaining cheese. Top with the remaining tortillas.
Wipe the skillet clean and melt the butter over medium-high heat. Carefully transfer 2 of the quesadillas to the skillet and cook, flipping once, until golden brown and melty, about 3 minutes per side; transfer to a cutting board. Cook the remaining 2 quesadillas. Cut each into 4 pieces and serve with salsa.