Heat a very large skillet over medium-high heat. Add the oil, then the mushrooms and shallots, season with salt and pepper, and sauté for 10 minutes until the liquid has evaporated and the mushrooms have turned nicely browned. Remove them to a bowl and stir in the parsley.
Wipe out the skillet and return it to medium-high heat. Add a half teaspoon of butter, and turn to coat the bottom of the pan. Place a tortilla in the pan. Sprinkle 2 tablespoons of the cheese over half of the quesadilla, and distribute ⅛ of the mushroom mixture over the cheese. Top that with another 2 tablespoons of the crumbled cheese. Flip the bare half of the tortilla over the filling, cover the pan, and sauté for about 2 minutes until the bottom is golden and the cheese has started to melt. Use a spatula to flip the half-moon quesadilla, and continue to cook, uncovered, until all of the cheese is melted and the underside is browned, 2 to 3 minutes.
Remove the quesadilla to a cutting board and let the quesadilla sit for a minute before you slice into 2 or 3 wedges. Repeat until all of the quesadillas are cooked. Serve with salsa or tapenade, as desired.
Notes
Other ingredients can be added along with the mushrooms. Try chopped olives, cooked broccoli or asparagus, tomatoes, artichoke hearts, cooked onions, roasted or sautéed bell peppers, or scallions. If you are not looking to make vegetarian quesadillas, you can also play with shredded cooked chicken, browned ground beef or turkey, crumbled bacon, or cooked shrimp.
When sauteeing mushrooms, they will release some liquid. You want to keep cooking them until that liquid evaporates, and then the mushrooms will be able to get nicely caramelized.
If your goat cheese is very creamy, you may want to spread a nice thick layer on the tortillas instead of trying to crumble it.