To make the compote, combine all of the ingredients and bring to a simmer over medium heat in a stainless-steel pot. Allow the compote to cook until the apples are tender. Remove from the heat and keep warm.
To make the omelet, heat a well-seasoned 8-inch sauté pan over medium-high heat. Meanwhile, in a stainless-steel bowl, whisk the eggs thoroughly. Season the eggs with a pinch of salt and pepper. Add the butter to the sauté pan and allow it to melt. Once the butter has stopped foaming, add the eggs. Using a fork or wooden spoon, stir the eggs while vigorously shaking the pan (as if making scrambled eggs).
Using the spoon, pull and hold the cooked eggs to one side of the pan and tilt the pan to allow the uncooked egg to gather on the other side. Flatten the pan and sprinkle the chives and cheese over the unset side of the eggs. Tilt the pan again toward the unset eggs and, using the fork or wooden spoon, start rolling the cooked layer of the eggs toward the pool of unset eggs—stopping once the roll reaches the largest part of the unset eggs. Straighten the pan and fold up the largest part of the unset eggs so that it meets the first roll to form the omelet.
Cover the pan with an inverted plate then carefully flip the omelet out onto the plate. Using a paring knife, make a slit lengthwise and fill the omelet with the apple compote and shredded Gouda. Season with salt and pepper to taste. Serve immediately.