Combine the Monterey Jack cheese and cheddar cheese in a medium bowl.
Sprinkle about 1½ tablespoons of the cheese mixture on one of the tortillas, followed by 1 tablespoon of the black beans, 4 or 5 spinach leaves, and then another 1½ tablespoons of the cheese mixture.
Top with a second tortilla to enclose the fillings.
Repeat with the remaining tortillas, cheese, black beans, and spinach.
Put 1 tablespoon of the olive oil in a large skillet over medium-high heat.
When it’s hot, add one of the quesadillas and cook until lightly browned on the bottom, 2 to 3 minutes.
Flip and cook until lightly browned on the other side, 2 to 3 minutes.
Cool on paper towels.
Repeat with the remaining quesadillas, adding the remaining tablespoon of olive oil to the pan when it becomes dry.
Cut the quesadillas in half or in wedges if desired, and serve warm with salsa.
