Arrange 4 tortillas on a flat work surface. Evenly divide the Pepper Jack, mushrooms, beef, and peppers on each tortilla taking care to leave a 1-inch border around the edge. Season to taste with salt and pepper. Using a spatula, spread 2 tablespoons cream cheese on each of the 4 remaining tortillas and set, cream cheese side down, on top, pressing gently to secure.
In a 9-inch skillet heat 1 tablespoon of the butter or oil over medium heat. Place one quesadilla in the pan, and cook until cheese begins to melt and tortilla crisps to a golden brown, about 5 minutes. Carefully flip the quesadilla and repeat on the other side. Transfer the hot quesadilla from the pan to a cutting board and cover with foil. Let rest for 2 minutes before cutting into wedges. Repeat with the remaining quesadillas. Serve immediately.