Roast the Vegetables: Preheat your broiler to high. Place the tomatillos, poblano peppers, jalapeƱos, and onion quarters on a baking sheet. Broil for 5-7 minutes, then flip and broil for another 5-7 minutes until the vegetables are softened and nicely charred in spots. Add the unpeeled garlic cloves to the pan during the last 5 minutes of roasting.
Blend the Sauce: Once cool enough to handle, peel the charred skins from the poblano peppers and squeeze the roasted garlic from their peels. Transfer all the roasted vegetables, garlic, fresh cilantro, and lime juice to a blender. Add 1/2 cup of the chicken broth. Blend until smooth. The sauce can be slightly chunky if you prefer.
Sear the Chicken: Pat the chicken thigh pieces dry with a paper towel and season generously with salt, pepper, cumin, and oregano. Heat 1 tbsp of olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add half of the chicken in a single layer and cook until browned on all sides, about 5-6 minutes. Remove to a plate and repeat with the remaining oil and chicken.
Simmer the Chili: Return all the browned chicken to the pot. Pour the blended chili verde sauce over the chicken. Add the remaining 1/2 cup of chicken broth to the blender, swirl to collect any remaining sauce, and add it to the pot. Stir everything to combine.
Finish and Serve: Bring the chili to a boil, then reduce the heat to low, cover, and let it simmer for at least 30 minutes, or up to an hour, for the flavors to meld and the chicken to become incredibly tender. If the sauce is too thick, add a splash more broth. Taste and adjust seasoning with salt and pepper. Serve hot with your favorite toppings.