Cook posole, covered in 1 quart unsalted water over medium low heat, stirring occasionally until soft but not mushy, about 3 hours for dried or 1 1/4 hour for frozen. Add more water as needed.
Meanwhile, trim one inch off the bottom stems of parsley and cilantro and discard. Roughly chop remaining leaves and stems. Puree the poblanos, onion, garlic, parsley, cilantro, and 1/2 cup broth in a blender until smooth.
When posole is just starting to split open, pour chile mixture over it and stir to combine. Add enough of the remaining stock to make posole soupy. Continue to simmer, uncovered, until posole is tender, adding more stock if necessary, about 45 minutes. Stir in lime juice. Serve with lime wedges, radishes, cilantro, and sour cream if desired.