Peel and chop onions and place in a cold saucepan. Add butter and heat covered over medium heat. The onions will soon begin to release moisture and eventually cook in their own juices, resulting in a highly concentrated flavor. Stir with a wooden spoon every 5 minutes for 30 - 40 minutes. Be careful so they don’t burn. (If the onions begin to become dry, you can add the vinegar a bit earlier.)
Remove the lid, add the vinegar and rosemary. Increase the heat and bring to a boil. Cook to reduce, while continuously stirring, for about 15 minutes. Set aside.
Beat 3 eggs, 2 tablespoons milk, salt and pepper (season to taste) per omelette. Heat 1 tablespoon butter in a skillet over medium heat. Add egg
mixture and place 4 cheese slices on top.
When the omelette starts to set around the edges but is still creamy on the surface, it’s ready to be served. Top with a generous spoon of caramelized onions and fold
the omelette in half. Garnish with chives, salt and pepper.