In a large skillet over medium-high heat, warm the oil. Add the mushrooms and spread out into an even layer. Let cook, undisturbed, for 5 minutes. Give them a stir and cook for 2-3 more minutes, or until deeply browned in spots.
Add the garlic and ½ teaspoon salt and cook for 30 seconds. Stir in the beans and remaining ½ teaspoon salt and cook for 1 more minute. Transfer to a bowl.
In a small bowl, whisk together all of the taco sauce ingredients until combined.
Lay out your tortillas and spread about ½ cup of cheese on half of the tortilla. Spread about ¼ of the mushroom bean mixture (about 1 cup) on top of the cheese, then sprinkle another ¼ cup cheese on top. Fold the naked half over the filled half into a half moon shape. Repeat with remaining 3 tortillas, cheese, and filling.
Wipe out the skillet and place it over medium heat. Add a thin layer of oil and once the oil is shimmering, swirl it to coat the pan or use a brush to spread it around. Place one or two tortillas in the pan, gently press with your spatula, then cook for 3 minutes or until golden brown. Flip and repeat.
Repeat this until all of your quesadillas are cooked. Slice them into triangles and serve sprinkled with some cilantro (if using) and taco sauce on the side.