Prep the potatoes: Scrub the potatoes, peel them if you like (I don’t), then slice them lengthwise into slabs, 3/8 inch thick.
Cut the slabs lengthwise into 3/8-inch batons.
Pour the white vermouth into a deep, 4-quart, nonreactive saucepan over high heat and boil until the liquid is almost completely evaporated; add the cream and boil the liquid again until it is reduced by half. Stir if necessary to prevent it from boiling over. Meanwhile, split the fennel bulb in half lengthwise, remove the core, trim off the leafy end and reserve for another use. Slice the bulb into ¼-inch slices.
Add the sliced fennel and garlic to the reduced cream, bring it back to a boil; turn off the heat and set aside.Give the potatoes their first fry: Preheat the oil in a deep fryer (or tall, deep sided stock pot) to 325 degrees. Line a sheet pan with paper towels to absorb the oil from the cooked potatoes.
Pour the potatoes from their water filled bowl into a colander then pat them completely dry with paper towels. Fry the potatoes in several batches to avoid crowding the oil, stirring them to distribute evenly in the oil until they just begin to turn golden, about 4 to 5 minutes. Move the par-cooked potatoes to the paper towel lined sheet pan and let them come to room temperature, at least 10 minutes and up to an hour before serving time. Raise the temperature of the oil to 375 degrees.Cook the mussels: Bring the cream sauce back to a boil over medium high heat; add the mussels, cover the pan and cook, shaking the pan once or twice until they begin to open; about 2 minutes.
Remove the lid and shake the pan again.
Pour in the wine, shake and cook for another 2 minutes or so until the wine reduces some. Cover the pan, remove from heat and set aside.Give the potatoes their second fry: Working with about half the potatoes at a time, fry in the 375 degree oil until golden brown and crisp, about 4 minutes.
Drain them on a paper-towel lined baking sheet; toss with kosher salt.
Pour the mussels and all the broth into a large bowl; garnish with chives.
Serve in individual bowls with the frites on the side.
