01 - In a mixing bowl, whisk together the eggs, milk, salt, and black pepper until well combined and slightly frothy.
02 - Heat a non-stick skillet over medium heat. Add the chopped red onion, bell pepper, and jalapeño; sauté for 2 to 3 minutes until softened. Stir in the diced tomato and black beans, then cook for an additional minute. Transfer the vegetable mixture to a bowl and wipe the skillet clean if needed.
03 - Add a light coating of oil or butter to the skillet. Pour in the egg mixture, swirling the pan to evenly coat the surface. Cook undisturbed for 1 to 2 minutes until the bottom is mostly set.
04 - Sprinkle the sautéed vegetables evenly over one half of the omelette. Top with the shredded cheese and half of the chopped cilantro.
05 - Using a spatula, gently fold the omelette in half to enclose the filling. Cook for another 1 to 2 minutes until the cheese is fully melted and the eggs are cooked through.
06 - Slide the omelette onto a serving plate. Garnish with the remaining cilantro, avocado slices, salsa, and lime wedges. Serve immediately.