Cook the mushrooms: Heat 1 tablespoon olive oil with 1 tablespoon vegan butter in a large skillet over medium-high heat. Add the mushrooms in a single layer and cook undisturbed for about 3 minutes, or until browned on the bottom. Stir the mushrooms and cook for about 3 minutes more, or until evenly browned.
Add the greens: Add the garlic and cook for another 30 seconds. Add the spinach and cook for about 3 minutes more, or until just wilted. Transfer to a medium dish.
Make the cashew cheese sauce: Place the nutritional yeast, cornstarch, mellow miso, onion powder, garlic powder, ground mustard seed, cashews, lemon juice, and 1 ¼ cups water into a high-speed blender. Blend on high until smooth.
Cook the cheese sauce: Pour the cashew sauce into the skillet and cook over medium-high heat for about 7 minutes, or until thickened, stirring constantly. Set aside.
Make the quesadillas: Heat 2 teaspoons olive oil in a large skillet over medium-high heat. Place one tortilla into the skillet. Spread one quarter of the cashew cheese sauce onto the tortilla. Add one quarter of the mushroom-spinach filling to one half of the tortilla. Fold over the other half to cover the filling. Heat for about 3 minutes, or until golden brown on the bottom. Flip and cook the other side for about 3 minutes more, or until golden brown.