Rinse and scrub the mussels under cool running water. Remove any "beards" by pulling the fiber down toward the hinge end. (I keep a clean pair of needle-nose piers around for this.) 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 stock, miso paste, fish sauce, and dark brown sugar to a container with a lid. Stir with a fork to break up the miso, then close the lid and shake vigorously to combine.
Heat the vegetable oil in an 11-inch straight sided skillet over medium-high heat until shimmering, about 1 minute. Add the shishito peppers and cook for 2 minutes or until fragrant and just beginning to brown. Add the ginger and cook for another minute, then stir in the broth mixture and the beer. Bring to a simmer.
Add the mussels and cover with a lid. Cook, shaking the pan occasionally, until the mussels have opened, about 2 minutes. (There are usually one or two that refuse to open no matter how long you cook them. Just discard as you find them.)
Transfer the mussels to a large serving bowl with a slotted spoon. Whisk the garlic chili sauce and lime juice into the broth and cook for 1 minute.
Whisk in the butter until melted and thoroughly integrated, Pour over the mussels, followed by the chopped cilantro. Serve immediately, mopping up the broth with the baguette.
*Fresh- and by that I mean- live. Mussels are available year-round and although they're considered to be at their peak from October to March, I eat a lot more in the summer than I do in the winter. When shopping, you'll find mesh bags, on ice, in the fresh seafood counter of quality markets. The shells should be shiny and most should be closed. It's ok if they're a little bit open as long as they close within a minute when tapped. If they're gaping wide, like they're yawning, pass. To store at home, I usually stash their mesh bag in a wide bowl over ice, topped by a moist towel. Don't wash them until right before cooking as fresh water will kill them. I strongly advise they be cooked within 24 hours of purchase...sooner is better.