Step 1: Begin by roasting your poblano peppers. You can do this under a broiler, on a grill, or over an open flame until the skins are charred and blistered. Place them in a bowl, cover tightly with plastic wrap, and let them steam for about 10-15 minutes. This makes the skins easy to peel.
Step 2: Once cooled, peel the charred skins from the poblanos, remove the stems and seeds, then dice them finely. Set aside.
Step 3: In a large pot or Dutch oven, heat olive oil over medium heat. Add diced onion and cook until softened, about 5 minutes. Stir in minced garlic and cook for another minute until fragrant.
Step 4: Add the diced poblano peppers, cumin, and smoked paprika to the pot. Sauté for 2-3 minutes, allowing the spices to bloom and the poblano flavors to deepen.
Step 5: Pour in the chicken broth and bring the mixture to a simmer. Reduce heat and let it gently simmer for 10 minutes to allow the flavors to meld.
Step 6: Stir in the cream cheese until it's fully melted and incorporated into the soup, creating a smooth base. Then, slowly whisk in the heavy cream.
Step 7: Add the cooked, shredded chicken to the pot. Gently stir in the shredded Monterey Jack cheese until it's melted and creates gooey ribbons throughout the soup.
Step 8: Season with salt and black pepper to taste. Finish with a squeeze of fresh lime juice and a generous sprinkle of fresh cilantro before serving.