Prepare chili oil: in a blender, combine 1 cup olive oil, the New Mexico chilies, the chile de arbole and ancho powder. Puree. Strain through a fine strainer, and reserve. (Recipe to this point can be made a day ahead.)
04To serve, preheat the oven to 425 degrees. Place the mushrooms in a large baking dish, drizzle with 1/2 cup chili oil and top with cheese slices. Bake until hot, 5 to 8 minutes. Remove from oven, garnish with thyme, and drizzle with a little more chili oil (any remainder will keep for three days). Serve hot.