Remove the husks from the tomatillos, rinse them, then roughly chop them. Chop the red onions, and jalapeno peppers (remove the seeds). Cut the garlic cloves into quarters.
Line a baking sheet with parchment paper. Place the chopped veggies on the sheet, drizzle with enough olive oil to coat, and sprinkle with kosher salt. Roast until soft and slightly browned, about 20 minutes, stirring occasionally.
Meanwhile, drain and rinse the hominy and black beans. Also, prepare the garnishes: thinly slice the radishes and green onions, and chop the cilantro.
Add the roasted vegetables to a large pot. Add the vegetable broth, and use an immersion blender to puree until fully blended (alternatively, transfer the hot soup to a blender, then blend and return it to the pot).
Add 3 sprigs of cilantro and the hominy and black beans. Bring the mixture to a boil, then lower the heat and simmer for 15 minutes. Season with kosher salt and fresh ground black pepper.
Remove the cilantro sprigs. Pour into bowls, and garnish with green onions, radishes, cilantro, and fresh-squeezed lime juice.