Season the salmon lightly with salt. Brush the top of the salmon with a little melted butter and immediately sprinkle evenly with crushed pepper and chopped ginger. Drizzle the remaining butter over the salmon. If you wait too long, the butter will harden and the mixture won't stick. Refrigerate, covered, until needed.
Preheat the oven to 500 degrees F.
Prepare the sauce: In a small saute pan, melt 2 tablespoons of the butter until foamy. Over medium heat, saute the shallots and garlic until the shallots are translucent, 2 to 3 minutes. Stir in the tomato and cook 1 to 2 minutes longer. Pour in the wine and vinegar, turn up the heat a little, and reduce until 1/2 cup remains. Pour in the chicken stock and reduce by half. Strain into a clean pan. Finish sauce by whisking in the remaining 4 tablespoons butter and season, to taste, with salt and pepper. Keep warm.
Brush some olive oil over a baking tray large enough to hold the salmon slices in 1 layer and arrange the salmon on the pan. Roast until medium, about 10 minutes. The salmon should be cooked on the outside, but still moist and slightly underdone on the inside.
Divide the sauce among 4 warm dinner plates. Spoon equal amounts of celery root puree in the center of each plate and place 1 piece of salmon on top.
Place the chicken bones in a 6-quart pot, pour water over to cover and bring to a rolling boil. Skim off the scum that collects on the top. Add the remaining ingredients, lower the flame, and simmer about 2 hours, skimming as necessary. Strain into a clean bowl and cool. Refrigerate in a container for up to 3 days, discarding the hardened layer of fat before using or freezing.
Yield: about 2 quarts
Celery Root Puree:
Place the celery root and potato in a medium saucepan and cover with cold water. Season with salt and cook until soft, 15 to 20 minutes. Drain the water and return the celery and potato to the pan. Pour in the cream and simmer over medium heat, stirring occasionally to prevent sticking, until thickened and most of the cream is absorbed, about 10 minutes.
Remove from the heat, stir in the butter, and season with salt and pepper, to taste. Puree in a food mill, return to the pan, and keep warm over simmering water. Use as needed.