First prepare your beer-pickled jalapeños. In a medium saucepan, add the vinegar, beer, honey, salt and garlic. Heat over medium heat for 3-4 minutes or until the honey and salt are dissolved and the alcohol from the beer has cooked out. Remove from heat and add the jalapeños. Set aside to pickle for 30-35 minutes. The jalapeños will store in an airtight container in the refrigerator for two weeks.
In a large saute pan, melt a tablespoon or butter over medium heat. Add the onions, season with salt and black pepper, and saute for about 8-10 minutes or until tender and brown. Remove the onions from the pan. To the pan, add another tablespoon of unsalted butter. Add the mushrooms, and cook until brown, about 6-8 minutes. Toss the onions back in with the mushrooms. Remove from heat and set aside until you are ready to assemble your quesadillas.
Preheat a griddle pan or large non-stick skillet over medium heat. Add a tablespoon or so of butter, and melt. Place one flour tortilla on the pan. Top with a couple of tablespoons of cheese, onion and mushroom mixture, beer-pickled jalapeños, and turkey. Top with a couple more tablespoons of cheese and place another flour tortilla on top. Press down. Flip the quesadilla carefully when the bottom is just a little brown, and cook for 2-3 minutes on the opposite side. Remove from heat, and repeat these steps 3 more times.
Cut into triangles and garnish with sour cream and cilantro. Serve with dipping sauces of choice and enjoy!