Combine the Dijon mustard, shallot and red wine vinegar in a large bowl. Season with salt and pepper. Slowly whisk in the olive oil and 3 tablespoons chives. Set aside.
Melt 2 tablespoons butter in a 12-inch nonstick pan over medium heat. When the butter starts to bubble, add the eggs and the cold cubed butter, and begin to stir with a rubber spatula. Continue to stir until the eggs are half set, then reduce the heat to low.
When the omelette is soft, sprinkle with salt and pepper, and top with the cheese on one side. Let the omelette sit over low heat, without touching it, for 1 minute, to let the heat of the pan set the eggs. Roll the omelette over the cheese so it forms a jelly roll shape, then roll the whole omelette onto a large dish. Lightly spread the remaining tablespoon butter over the omelette to coat. Toss the lettuce with the vinaigrette and serve alongside the omelette. Garnish the omelette with the remaining tablespoon chives.