Set an oven rack close to the broiler heat source while still able to fit the baking sheet between the rack and the broiler (3 to 4 inches from heat). Set broiler to high.
Mix oil, chili powder, 2 teaspoons salt, and 1 teaspoon black pepper in a small bowl. Put the tomatillos, poblano peppers, onion, and garlic on a baking sheet lined with aluminum foil; rub all over with the oil-chili powder mixture. Broil, tossing once, until everything is golden brown and can be easily pierced with a fork, 13 to 15 minutes. (It's OK if the vegetables get a bit charred).
Transfer the contents of the baking sheet to a blender and puree until completely smooth. Pour into a 5- to 6-quart stock pot or Dutch oven and add stock. Cover and bring to boiling over high. Reduce heat to a simmer and add the chicken, hominy, and beans. Cook until everything is heated through, 5 to 7 minutes more. Stir in the lime juice, sugar, 1 teaspoon salt and 1/4 teaspoon black pepper.
Serve with queso, radishes, cilantro, Quick-Pickled Red Onions, and/or lime wedges.