The mushroom mixture: Heat the oven to 350 degrees F. In a large skillet (preferably nonstick), heat the oil over medium-high. Add the sliced onion and cook, stirring frequently, until softening and beginning to brown, about 5 minutes. Add the mushrooms and stir nearly constantly until they have softened and any juice they release has evaporated, about 5 minutes longer. Stir in the sliced chiles, then taste and season with salt, usually about 1/4 teaspoon. Transfer the mixture to a 9- or 10-inch shallow baking dish, Mexican cazuela or pie plate.

Finishing the queso fundido: Very lightly dampen a clean kitchen towel. Check the tortillas to make sure none are stuck together. Wrap them in the towel, then in foil, sealing the edges tightly. Place in the oven and set the timer for 7 minutes. When the timer goes off, stir the shredded cheese into the warm mushroom mixture. Set in the oven alongside the tortillas and bake until the cheese is just melted, about 5 minutes longer. Sprinkle with the thyme or epazote and black pepper and serve without a moment's hesitation, accompanied by warm tortillas.