How to Make It
Heat 1 tablespoon butter in an 8 1/2-inch nonstick skillet over medium-high. Add onion, and spread in an even layer. Cook, undisturbed, until onions begin to brown, about 4 minutes. Add 2 tablespoons water and 1/8 teaspoon salt; cook, stirring constantly, until onions are golden brown and softened, 8 to 10 minutes. Transfer onions to a bowl. Wipe skillet clean.
Using a fork, stir together eggs, cream, and remaining 1/8 teaspoon salt in a medium bowl until well combined, being careful not to incorporate too much air.
Heat same skillet over medium-high. Add 1 tablespoon butter (it should melt almost immediately); swirl to evenly coat bottom of skillet. Just as butter stops foaming but before it begins to brown, add egg mixture all at once. Cook, stirring constantly with a heatproof rubber spatula, until mixture thickens slightly into very soft scrambled eggs with thin ribbons of set curd running throughout, 10 to 20 seconds. Gently shake skillet to form an even sheet of eggs on bottom of skillet.
Reduce heat to low. Crumble cheese over eggs. Sprinkle 1 tablespoon caramelized onions in a straight line separating the upper third from the bottom two-thirds of omelet. (Reserve remaining onion for another use.) Cook omelet, undisturbed, until filling heats and cheese melts, about 1 minute. Using spatula and starting with upper third side, gently fold omelet over, just covering filling. Starting at top of folded side, roll up entire omelet. Turn off heat, and let stand 10 seconds. Transfer to a warm plate, and rub hot omelet with remaining 1 teaspoon butter. Sprinkle with chives, and add salt to taste. Serve immediately.