Place the salmon fillets on a board and working with a very sharp knife that's held almost parallel to the board, cut them into 12 thin slices. Divide the sea salt evenly over the slices of salmon and reserve.
Place 1 tablespoon of the butter in a small, heavy saucepan. Melt over medium-low heat. Add the shallots and cook for 1 minute. Cut out the white bulb of the fennel and cut it into tiny dice. You need 1 cup of diced fennel; if any of the bulb is left over, save it for another use. Reserve 4 teaspoons of the frilly green leaf at the other end of the fennel. Add the cup of diced fennel to the sauce pan and stir well to coat with the butter. Add the wine and the vinegar to the sauce pan. Reduce over high heat, until just a small amount of liquid remains ( a teaspoon or so). Moving the pan on and off the heat as required, start incorporating the remaining 10 tablespoons of butter, 1 tablespoon at a time, beating with a broad wooden spoon. The sauce should be smooth, creamy, and medium-thick. Taste for seasoning.
While finishing the sauce, steam the salmon slices: Place them on a buttered steamer rack and cook them over barely simmering water until just done (approximately 3-6 minutes). They should retain just a trace of red at the center. Remove and divide among 4 dinner plates stacking them on steamed fennel pieces. Pour the sauce around the fillets on each plate and garnish with the reserved fennel leaves.