Heat a large skillet over medium heat. Add 1 teaspoon butter and let it melt, coating the pan.
Crack 1 large egg into the skillet. Sprinkle with a pinch of salt and pepper. Cook until the egg white is set and opaque.
Gently flip the egg with a spatula. Immediately sprinkle half of the 1 ounce freshly grated sharp white cheddar over the cooked side of the egg. Center the 14-inch flour tortilla on top of the egg.
Cook for 1 minute to warm the tortilla.
Using a wide spatula (or two spatulas), carefully flip the tortilla-and-egg assembly so the tortilla is now against the pan and the egg is on top.
Sprinkle the remaining grated cheddar over the egg, cover the skillet with a lid, and cook for 1 minute to melt the cheese.
Slide the tortilla egg onto a plate and top with the sliced tomatoes, green onions, avocado, jalapeño peppers, and fresh cilantro. Serve.