Rinse and scrub the mussels under cool running water. Remove any "beards" by pulling the fiber down toward the hinge end. Discard the beards and any open mussels that do not close within 30 seconds after a hard tap. Set the mussels aside while you prepare the broth.
Add the chicken broth, brown miso paste, fish sauce, and dark brown sugar to a container with a lid. Stir with a fork, then close the lid and shake vigorously to combine.
Heat the vegetable oil in a large saucepan over medium-high heat until shimmering. Add the shishito peppers and cook until fragrant and just beginning to brown, about 1 minute. Add the ginger and garlic and cook for another minute. Stir in the chicken broth mixture to combine. Pour in 1 bottle of beer and bring to a simmer. Add the mussels, cover, and cook until most of the mussels have opened, about 2 minutes.
Use a slotted spoon to remove the mussels and place in two serving bowls. Stir the harissa and juice from 2 limes into the broth. Pour the broth over the mussels and finish with the cilantro. Serve with the lime quarters, leftover beer, and some crusty bread.