1. Preheat oven to 350 degrees.

2. Place chicken breast, skin side up, on a rimmed baking sheet; drizzle with olive oil and generously season with salt and pepper. Transfer to oven and bake for 40 minutes. (Chicken may not be cooked through.)

3. Remove from oven and let stand until cool enough to handle. Using your fingers or a fork, remove meat from bones and shred into large pieces; set meat aside, discarding bones.

4. Preheat broiler. Place tomatillos on a rimmed baking sheet. Place chiles, garlic, onion, and poblano peppers on another rimmed baking sheet. Transfer both baking sheets (one at a time, if necessary) to broiler and broil, turning once, until vegetables are charred and softened, about 10 minutes.

5. Transfer poblanos to a small bowl and cover with plastic wrap; let steam for 10 to 15 minutes. Let remaining vegetables cool slightly and peel garlic. Uncover poblanos and, using a paper towel, remove skins. Remove seeds and stems and coarsely chop. Transfer chopped poblanos to the jar of a blender along with tomatillos, chiles, garlic, onion, cilantro, cumin, and lime juice; blend until slightly smooth but a few chunks remain and season with salt and pepper.

6. Transfer pureed tomatillo mixture to a large saucepan and add reserved chicken; place over medium heat and simmer until chicken is no longer pink and completely warmed through. Season with salt and pepper and add more lime juice, if desired. Keep warm until ready to use.