Season the chicken on both sides with salt and pepper.
Heat oil in a lidded Dutch oven over medium-high heat, then brown the chicken well on both sides. 7 or 8 minutes. You will need to cook in batches so you don’t overcrowd the pot.
Reduce heat to medium, then add onions, stirring often until softened, but not browned. 3-5 minutes. Add broth and scrape up any brown bits on the bottom or sides of the pot, then add the poblanos, garlic, tomatillos, cilantro, and cumin. Bring to a boil then reduce heat to a simmer. Return the chicken and accumulated juices to the pot, nestling it amongst the sauce, then cover and simmer 20-30 minutes until cooked through and tender. White meat will cook sooner than dark, so if you’re using both, pull the white pieces out first and allow the dark to finish up.
Remove the chicken to a plate, tenting it with foil, then increase the heat to medium-high. Add the lime juice and cook, allowing the flavors to marry and the sauce to thicken slightly. Adjust seasoning.
Pour sauce over chicken and garnish with whole cilantro leaves. Serve with warm flour tortillas.