1 cup cooked, shredded chicken
1 (10 ounce) can red enchilada sauce, divided
2 tablespoons mild green chiles, canned or fresh
1/4 teaspoon chili powder
4 large eggs, separated
1 tablespoon heavy cream or milk
1 tablespoon unsalted butter
1/4 cup grated pepper jack cheese, divided
1/4 cup grated cheddar cheese, divided
3 tablespoons crushed tortilla chips
Kosher salt and freshly ground black pepper
Salsa, sour cream, and salsa verde, for serving
Combine the chicken, 3/4 cup enchilada sauce, green chiles, and chili powder in a medium skillet and cook on moderate heat until warmed through. Season with salt and pepper. Reduce heat to low and keep warm.
In a large mixing bowl, whisk the egg yolks and cream along with a generous pinch of salt and pepper. In the bowl of a stand mixer (or with an electric hand mixer), beat the egg whites on high speed until stiff peaks form, 4 to 5 minutes.
Melt the butter in a 12-inch nonstick skillet over low to medium-low heat (a 10-inch skillet will work but the omelet will be on the thicker side). Gently fold the egg whites into the yolks until completely combined. Pour the mixture into the skillet and allow to cook until almost completely set. Sprinkle 2 tablespoons of the pepper jack and 2 tablespoons of the cheddar cheese along with the tortilla chips over the surface of the omelette. Transfer the skillet to the broiler to melt the cheese, 1 to 2 minutes.
Spread a layer of the warm chicken mixture down the center of the omelet. Run a silicone spatula around the edges to loosen it from the skillet. Fold it in half and transfer to a plate. Drizzle with a few tablespoons enchilada sauce and the remaining cheese. Serve with salsa, sour cream, and salsa verde on the side.