Preheat the oven to 350 degrees F. Line a baking sheet with aluminum foil.
Melt 2 tablespoons of the butter in a medium skillet over medium-high heat. Add the mushrooms, shake the pan to distribute them evenly in the pan, and then cook without disturbing until the mushrooms begin to brown, about 5 minutes. Do not add salt. Add the remaining 1 tablespoon butter along the side of the skillet if the skillet seems very dry. Toss the mushrooms, and continue to cook until nicely browned, about 2 to 3 minutes more. Season mushrooms with the chili powder and salt and cook for 1 minute. Stir in the chopped cilantro and set aside.
Toast the tortillas directly on a medium gas burner, turning occasionally until slightly blistered on each side. (If you have an electric burner, do this in a dry cast-iron skillet.) Place the tortillas on the prepared baking sheet. Spread the jack cheese on one half of each tortilla, top with some of the mushrooms, and then sprinkle with the feta. Fold the other half of the tortilla over the filling, to make a half-moon shape. Lay another sheet of foil over the quesadillas. Bake until the cheese melts, about 12 minutes. Serve whole or sliced into wedges, with optional garnishes.