Season chicken with salt and pepper. Heat olive oil over medium-high heat in a large Dutch oven or pot. Add chicken and brown on both sides, about 3-4 minutes per side. Remove and set aside (it won't be fully cooked yet).
In the same pot, add onion and poblano peppers. Sauté until softened, about 5 minutes. Add garlic, cumin, oregano, coriander, and cayenne; cook until fragrant, about 1 minute.
Pour in the chicken broth, scraping up any browned bits from the bottom of the pot. Return the chicken to the pot, bring it to a simmer, and reduce the heat to medium-low. Cover and cook until tender, about 15-20 minutes.
Remove chicken and shred with two forks. Return shredded chicken to the pot.
Add white beans and diced green chiles. Simmer uncovered for 20-30 minutes, allowing the chili to thicken slightly.
Stir in lime juice and heavy cream. Simmer gently for 5 more minutes. Add cilantro and adjust seasonings to taste.
Serve hot with your choice of toppings.