In a small bowl, add the Greek yogurt and goat cheese and a pinch of salt and pepper. Whisk together and set aside. In another small bowl, whisk eggs together lightly until its a uniform color. Add 1/2 the dill.
Heat an 8-inch nonstick pan over medium-high heat, add 1 tablespoon butter and swirl to coat the pan. Add shallots and a pinch of salt and sauté for 2-3 minutes until they start to sizzle, then lower the heat to to low and cook another 8-10 minutes until shallots are soft and translucent but not brown. Add the other tablespoon butter, let melt and coat the pan, then pour in eggs. I like to leave the heat on medium- low as to not brown the edges. Shake the pan for a few seconds. As the eggs begin to set, tilt the pan and lift the set eggs so the liquid flows underneath. Move the spatula along the edge of the omelette to shape the omelette and make sure it isn’t sticking.
Once the omelette is almost set (This takes a few minutes. I like mine a little runny in the center) add the lox and yogurt goat cheese sauce on half of the omelette and fold over the other half.
Slide onto a plate and eat! Garnish with chives and more dill first. I serve mine with some greens, fruit and a mimosa. Duh.