How to Make It
In a medium saucepan, cover the sunchokes with cold water and bring to a boil. Add salt and cook over moderately high heat until tender, 15 to 18 minutes. Drain, reserving 1/4 cup of the cooking liquid. Transfer the sunchokes to a food processor and puree until smooth; if necessary, add a few tablespoons of the cooking liquid. Return the puree to the saucepan, stir in 2 tablespoons of the butter and season with salt and pepper.
In a medium nonreactive saucepan, boil the balsamic vinegar over high heat until reduced to 2 tablespoons, about 5 minutes. Add the chicken stock.
In a second medium saucepan of boiling salted water, cook the fennel wedges over moderately high heat until just tender, about 15 minutes. Drain the fennel reserving 1/2 cup of the cooking liquid. Rinse the fennel in cold water, then drain and transfer to a shallow baking dish with the reserved fennel cooking liquid.
Preheat the oven to 450°. Cut 1 tablespoon of the butter into small bits and dot the fennel with them. Season lightly with salt and pepper. Bake the fennel for about 10 minutes, or until tender and heated through; keep very warm. Leave the oven on.
Meanwhile, heat the oil in a large cast-iron skillet until shimmering. Dredge the skin side of the salmon in flour, tapping off any excess. Add the salmon to the skillet, skin side down, and season with salt and pepper. Cook over moderately low heat until the skin is just browned and the flesh near the skin turns opaque, about 5 minutes. Transfer the skillet to the oven and roast the salmon for about 4 minutes, or until barely cooked through. Transfer to a large plate, skin side up, and let rest for about 5 minutes.
Gently rewarm the sunchoke puree over low heat. Rewarm the balsamic glaze over moderate heat. Whisk the remaining 5 tablespoons of butter into the glaze, 1 tablespoon at a time; don't let the sauce boil.
Spoon the sunchoke puree and the fennel onto 6 large plates. Set the salmon, skin side up, on the puree. Spoon the balsamic sauce around the fish and garnish with the fennel fronds.