How to Make It
In the bowl of a stand mixer, combine the 1 cup of flour and the salt. Using your hands, rub the bacon fat and vegetable shortening into the flour until the mixture is crumbly. Using the dough hook, slowly mix in the water at medium speed until a ball forms, adding more flour if necessary; the dough will be moist but should start to pull away from the side of the bowl. Continue kneading until the dough is smooth, shiny and elastic, about 10 minutes. Cover the bowl with plastic wrap; let rest for 45 minutes.
Scoop slightly rounded tablespoonfuls of the dough and roll between pieces of parchment paper into 4-inch rounds. Heat a cast-iron skillet and cook the tortillas in batches over moderately high heat, turning once, until lightly browned, about 1 minute total. Transfer to a kitchen towel and keep warm.
In a medium skillet, melt the butter. Add the mushrooms and thyme and cook over moderately high heat, undisturbed, until lightly browned, about 2 minutes. Season the mushrooms with salt and cook, tossing occasionally, until browned and crisp, about 6 minutes. Transfer to a paper towel-lined plate; discard the thyme sprigs.
Preheat the broiler. In a small cast-iron skillet or ovenproof dish, broil the cheese 8 inches from the heat until just melted, about 1 minute. Top the cheese with the mushrooms, chile and cilantro and serve immediately with the warm tortillas.
If bacon fat is unavailable, double the amount of vegetable shortening.