Preheat the oven to 400 degrees Fahrenheit. Heat a medium skillet over medium heat and add the oil.
Add the sliced mushrooms and sauté until mostly tender and some liquid has released and evaporated, about 5 minutes. Stir in the spinach and a pinch of salt and pepper and sauté until the spinach is wilted, about 2 minutes. Remove from heat and let cool slightly.
Assemble the quesadillas: Place some of the shredded cheese on one half of each tortilla. Top the cheese with some of the sautéed mushrooms and spinach and fold the other half of the quesadilla over the filling. Place on a large baking sheet. Repeat with the remaining tortillas, cheese, and sautéed veggies.
Bake in the preheated oven for about 10 minutes, flipping the quesadillas halfway through, until the cheese is melted and the outside of the tortillas are crisp.