Roast the poblano peppers over an open flame or under the broiler until the skin blackens all over.
Place roasted peppers in a bowl, cover with plastic wrap, and let them steam for 10 minutes.
Peel off the charred skin, remove the seeds, and chop the peppers. Set aside enough for the omelette and the sauce.
In a blender, combine one roasted poblano, garlic, crema, lime juice, cumin, and cilantro. Blend until smooth and set the sauce aside.
Heat a small saucepan and cook the diced onion and bell pepper until soft, about 4 minutes.
Add the shredded beef to the pan just to warm it through, then remove from heat.
In a mixing bowl, whisk the eggs with milk, salt, and pepper until fully combined.
Heat butter in a nonstick skillet over medium heat until melted.
Pour in the eggs and let them cook undisturbed until the edges start to set.
When the center is still slightly wet, add the beef mixture, the remaining chopped poblano, and half the cheese.
Fold the omelette in half using a rubber spatula.
Cook for another minute so the cheese melts fully and the eggs finish cooking through.
Slide the omelette onto a plate.
Top with sliced avocado, a generous drizzle of poblano cream sauce, and chopped serrano pepper.
Serve immediately while everything is warm and the cheese is still melty.
