In a medium bowl, whisk eggs, 2 tbsp.4 cold water and a pinch each salt and pepper until blended.
Heat an 8-inch, heavy nonstick skillet with low, sloping sides over medium heat until hot enough to sizzle a drop of water. Add butter and tilt the pan to coat. When the butter stops sizzling, add the eggs.
Cook until the eggs begin to set around the edges, 10 seconds. Using a spatula, lift and pull the set eggs toward the center, letting the unset eggs in the center move to the edges. Adjust the heat to prevent browning. Cook until barely set, 20 seconds.
Off heat, spoon 1/3 cup filling onto the omelet on the area closest to the handle. Carefully tilt the pan to roll the omelet and slide it onto a plate; nudge with a spatula if necessary.