Prepare the cornbread: Crumble the cornbread into a large bowl. Let it sit uncovered for 15–20 minutes so it dries slightly.
Sauté aromatics: In a skillet, sauté diced onion (and celery, if using) until soft and fragrant.
Add peppers: Stir in roasted poblano peppers or canned green chiles. Cook 1–2 minutes.
Combine mixture: Add shredded birria meat to the bowl with the cornbread. Pour in the sautéed vegetables.
Add seasoning: Stir in cumin, oregano, black pepper, and salt.
Mix in eggs: Add whisked eggs and toss to coat evenly.
Add consomé: Pour in 1 ½ cups warm consomé to start. Mix gently—add up to ½ cup more if needed until the mixture is moist but not soupy.
Transfer to pan: Spread the dressing mixture into a greased baking dish.
Bake: Bake at 375°F for 30–40 minutes, or until the top is golden and the center is set but still tender.
Serve: Scoop and serve warm with cilantro, lime, and extra consomé if desired.