Heat the oil in a large skillet over medium heat and sauté the tortilla pieces or chips until slightly crisp, 1 to 2 minutes. Add the tomato, onion, and chile and cook for 5 minutes. Pour in the eggs and mix with a spatula, scraping up the eggs as they cook. When the eggs are partially set, add the Cheddar. Cover. Toss a few more times until the cheese melts.