In a large skillet, saute the onion in butter on a medium-low heat for about 7 minutes, until it softens and starts to brown.
Add mushrooms, increase the heat to medium and cook for 5-7 minutes. Season with salt and pepper and set aside.
To assemble the quesadilla, spread 1/3 cup of mashed potatoes on one side of the tortilla and sprinkle with 1/4 cup of cheese. Spread 1/4 of the mushroom-onion mixture on top and sprinkle with 1/4 cup of cheese again.
Fold the tortilla in half and press slightly with your hand to bind the layers together. Repeat this step to make the remaining three Quesadillas.
Fry in a little butter for about 2 minutes on each side.