The mushroom filling.
In a medium skillet, heat the 2 tablespoons oil or lard over medium. Add the mushrooms, chiles, stir well and cover. Continue to stir every minute or so, replacing the cover each time, until the mushrooms have released a lot of juice, 4 to 5 minutes. Uncover and let briskly simmer until all the liquid has evaporated, about 3 minutes. Taste and season with salt, usually about 1/4 teaspoon. Add the cilantro now.
Forming and griddle-baking the quesadillas.
Heat a well-seasoned or nonstick griddle or heavy skillet over medium. Turn the oven on to its lowest setting. One by one, make the quesadillas: Lightly brush one side of each tortilla with oil, then lay it oiled side down on the hot griddle. Spread with about l/3 cup of the cheese, leaving a 1/2-inch border all around. Spoon about 3 tablespoons of the mushroom mixture down the center of the cheese-covered tortilla. When the cheese begins to melt (but before the tortilla begins to crisp), fold the tortilla in half along the line of the filling. Cook, flipping the quesadilla every minute or so, until the cheese is completely melted and the tortilla crisps, about 5 minutes in all. As each quesadilla is done, transfer it to a baking sheet and keep warm in the oven.
Serving the quesadillas. When all the quesadillas are done, set them out in a cloth-lined basket. Pass the salsa separately for each guest to spoon on