Heat a dry skillet over medium-high heat. Add tomatillos, poblano, and unpeeled garlic cloves. Cook, turning occasionally, until blistered and blackened in spots, about 8 minutes for the tomatillos and pepper, 5 minutes for the garlic. Let cool slightly, then peel the garlic.

Transfer charred vegetables to a blender with cilantro, lime juice, and 1/2 teaspoon salt. Blend until mostly smooth with some texture remaining. You should have about 1 1/2 cups.

Pat chicken dry and season with remaining 3/4 teaspoon salt. Heat olive oil in your large skillet over medium-high heat. Add chicken in a single layer and cook undisturbed until golden brown on the bottom, about 4 minutes. Flip and cook 2 minutes more (chicken will not be fully cooked). Transfer to a plate.

Add rice to the same skillet and stir constantly until grains turn opaque and smell nutty, about 2 minutes. This step prevents gummy rice.

Pour in salsa verde and chicken broth, scraping up any browned bits. Stir in black beans and return chicken with any accumulated juices to the pan, nestling pieces into the rice. Bring to a boil, then reduce heat to low, cover tightly, and simmer for 18 minutes until rice is tender and liquid is absorbed.

Remove from heat and let stand, covered, for 5 minutes. Fluff with a fork and taste for seasoning. Serve topped with cotija, crema, and extra cilantro.