Cut 1 tablespoon butter in half again. Cube remaining tablespoon butter into small dice, transfer to small bowl, and place in freezer while preparing eggs and skillet, at least 10 minutes. Meanwhile, place oil in 8-inch nonstick skillet and heat over low heat 10 minutes.
Crack 2 eggs into medium bowl and separate third egg; reserve white for another use and add yolk to bowl. Add 1/8 teaspoon salt and pinch of pepper. Break yolks with fork, then beat eggs at moderate pace, about 80 strokes, until yolks and whites are well combined. Stir in half of frozen butter cubes.
When skillet is fully heated, use paper towels to wipe out oil, leaving thin film on bottom and sides of skillet. Add ½ tablespoon reserved butter piece to skillet and heat until foaming subsides, 45 to 90 seconds. Swirl butter to coat skillet, add egg mixture, and increase heat to medium-high. Following photos above, use 2 chopsticks or wooden skewers to scramble eggs using quick circular motion to move around skillet, scraping cooked egg from side of skillet as you go, until eggs are almost cooked but still slightly runny, 45 to 90 seconds. Turn off heat (remove skillet from heat if using electric burner) and smooth eggs into even layer using rubber spatula. Sprinkle omelet with 1 tablespoon cheese and 2 teaspoons chives. Cover skillet with tight-fitting lid and let sit 1 minute for runnier omelet and 2 minutes for firmer omelet.
Heat skillet over low heat 20 seconds, uncover, and, using rubber spatula, loosen edges of omelet from skillet. Place folded square of paper towel onto warmed plate and slide omelet out of skillet onto paper towel so that omelet lies flat on plate and hangs about 1 inch off paper towel. Roll omelet into neat cylinder and set aside. Return skillet to low heat and heat 2 minutes before repeating instructions for second omelet starting with step 2. Serve.