Position a rack in high up oven and preheat oven to 400 degrees.
In a 12-inch ovenproof skillet, heat 2 tbsp. butter over medium-high heat until shimmering hot. Add sunchokes and salt; cook, flipping as needed, until well browned on both sides, 2-4 minutes per side.
Move skillet to oven and roast until sunchokes are tender, about 20 minutes. Transfer to a bowl and cover to keep warm.
Set skillet over medium heat, add shallot, cook and stir until softened and lightly browned, 1 to 2 minutes. Add wine and cook, stirring until wine has evaporated.
Reduce heat to low, add lemon juice, and then remaining butter. Stir in the parsley and tarragon. Return the sunchokes to pan and toss to reheat and coat with butter. Season to taste with salt and pepper before serving.