Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the chicken pieces and season lightly with salt and pepper. Cook, stirring occasionally, until browned on all sides, about 5-7 minutes. Browning the chicken adds a rich depth of flavor that's essential for a savory chili. If using rotisserie chicken, skip this step and add shredded chicken later.

Remove chicken from the pot and set aside. Reduce heat to medium. Add the chopped onion to the pot, scraping up any browned bits from the chicken. Sauté until the onion is softened and translucent, about 5 minutes. This creates the foundational aromatic base for your chili.

Stir in the minced garlic, diced poblano pepper, and canned green chiles. Cook for another 3-4 minutes, stirring frequently, until the peppers begin to soften and the garlic becomes fragrant. This step infuses the chili with a crucial layer of mild heat and earthy flavor.

Add the ground cumin, dried oregano, smoked paprika, and cayenne pepper (if using). Cook for 1 minute more, stirring constantly, allowing the spices to toast lightly. This 'blooming' of spices in oil intensifies their aroma and flavor, making your chili more vibrant.

Pour in the chicken broth and bring it to a simmer, scraping the bottom of the pot to release any remaining flavorful bits. Return the browned chicken (or add shredded rotisserie chicken) to the pot, along with the rinsed and drained Great Northern and cannellini beans. Stir well to combine.

Reduce the heat to low, cover the pot, and let the chili simmer gently for at least 20 minutes (or up to an hour for more developed flavors). This low and slow cooking allows the flavors to meld beautifully and the chicken to become incredibly tender.

Just before serving, stir in the heavy cream and fresh lime juice. Cook for another 2-3 minutes until heated through, but do not boil after adding cream, as it can curdle. The cream adds a luscious, velvety texture, and the lime brightens all the flavors, balancing the richness.

Stir in the fresh cilantro. Taste and adjust seasonings with additional salt and pepper as needed. Serve hot, garnished with extra cilantro, shredded cheese, sour cream, or avocado slices for a truly satisfying meal.