Dice the onion, mince the garlic, rinse and drain the black beans, shred the chicken, and measure the spices into a small bowl. Chop the cilantro, juice the lime, and shred the cheese; refrigerate until serving.
Set the broiler to high and place whole poblanos on a foil-lined baking sheet about 6 inches from the heat. Broil about 5 minutes per side, turning with tongs, until blackened and blistered all over (or char over a gas flame or in a dry cast iron skillet for 8 to 10 minutes).
Transfer the hot peppers to a bowl, cover tightly, and steam for 15 minutes. Rub off the blackened skin with your fingers (do not rinse), remove stems and seeds, and dice into small pieces.
Heat the olive oil in a Dutch oven over medium heat. Cook the onion for 4 to 5 minutes until soft and translucent, then add the garlic and stir for about 1 minute until fragrant.
Add the diced poblanos, corn, cumin, chili powder, Mexican oregano, salt, and pepper. Stir for 1 to 2 minutes until the spices smell warm and toasty.
Pour in the chicken broth and green enchilada sauce, scraping up any browned bits. Add the black beans and shredded chicken, bring to a gentle boil, then reduce the heat and simmer uncovered for 12 to 15 minutes.
Reduce the heat to low or turn it off. Slowly stir in the heavy cream and warm through for 2 to 3 minutes without boiling, then stir in the lime juice and cilantro and season to taste.
Ladle into bowls and top with shredded Monterey Jack, diced avocado, a dollop of sour cream, and crushed tortilla chips. Serve immediately.