Melt butter in 8-inch non-stick skillet over low heat, making sure butter is evenly spread across the bottom of the pan.
Meanwhile, in a cup, beat together the eggs and salsa. Pour the egg mixture into the pan.
Sprinkle cheese over the egg mixture and cover. Cook until done to your liking -- perhaps checking after 1 and then 2 minutes. Don't overcook; the omelet should still look quite moist and will continue to cook after you plate it.
Plate. Tip the pan and using a rubber spatula, carefully roll the omelet to the side of the pan onto the plate. (Or fold in half) Top with pickled onions. Serve hot, warm, at room temperature, or cold with extra salsa, more cheese, and fresh tomatoes (or avocado) if desired.