In an 8-inch nonstick pan, melt 1 tablespoon butter over medium heat until foaming. Add 1/4 of the crab mix to the pan and cook gently until the celery and onions are crisp but tender, approximately 10 minutes. While the crab mix is heating, in a separate large bowl, whip the eggs with a wire whisk until smooth. When the crab mix is hot, add 1/4 of the whipped eggs, about 1 cup.
Cook the omelet until the edges begin to curl. Gently pull the edges toward the center, allowing the eggs to cook through. Once the eggs have become firm, flip the omelet and cook until the other side is done, about 5 minutes. Flip the omelet out of the pan onto a dish and serve immediately.