Position a rack in the center of the oven and heat oven to 350°F.
In a small saucepan, combine the oils, chile powder, and garlic powder. Warm over medium-low heat until the oil becomes rust-colored and fragrant, about 4 minutes. Remove from heat and let cool slightly.
With a pastry brush, lightly coat both sides of the tortillas with the chile oil. Cut each tortilla into 6 triangles, arrange the triangles in a single layer on a baking sheet, and sprinkle the tops with salt. Bake until crisp, 15 to 18 minutes, flipping the chips halfway through baking.
Divide the tomatoes among six large salad plates, fanning them out in a circle. Tuck the cheese slices between the tomatoes and sprinkle with salt and pepper. Toss the arugula or mâche with 1 Tbs. of the chile oil, squeeze a little lime juice over, and season to taste with salt and pepper. Drizzle some of the remaining oil over the tomatoes and cheese (you won't need it all), mound the greens in the center of each plate, and arrange two tortilla chips to the side. Serve immediately.