Heat a large skillet over medium heat and add 2 tablespoons olive oil. Saute the asparagus, mushrooms and onions, about 4 minutes. Add the wine and some salt and pepper. Reduce the heat to low and simmer, about 5 minutes. Remove from the heat.
Wrap the tomatillos and chiles in aluminum foil and place over an open flame for about 5 minutes on each side. Transfer to a blender with the garlic clove and 1/3 cup water. Blend until smooth. Taste and season with salt and pepper.
Wrap the tortillas in a clean, damp dishcloth and place in the microwave for 2 minutes.
Fill each tortilla with a little jack cheese and about 1 tablespoon asparagus filling. Fold in half like a taco.
Heat a large saute pan over medium-high heat and add the remaining 2 tablespoons olive oil. Pan-fry the folded quesadillas in batches until golden and crispy, about 3 minutes on each side. Set aside on paper towels to drain any oil.
Serve each quesadilla with some goat cheese crumbled on top and garnish with fresh cilantro.
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.