In a small saucepan, melt 2 tablespoons of the butter. Add shallots and saute 1 minute. Season with a pinch of salt. Add lemon juice, wine, stock and cream. Simmer over medium heat for 3 to 4 minutes, until reduced to about 2 tablespoons.
Lower the heat and whisk in the remaining butter 1 tablespoon at a time, adding each piece when the previous one is almost melted.
Add hot sauce and season with salt and pepper to taste. Strain the sauce through a fine sieve, into a bowl. Cover to keep warm until ready to serve.
Mix together the mustard powder, salt, and pepper. Brush the top of the salmon fillets with the melted butter. Season both sides with the mustard mixture and set aside.
Put the soaked shingles in the oven, directly on the racks, for about 3 to 5 minutes until the wood is lightly browned on top. Carefully take the shingles out of the oven and place them on a heat-proof surface.
Turn on the oven broiler.
Immediately brush the shingles with a thin layer of olive oil and lay the salmon fillets, skinned side down, on the browned side of each shingle. Place the shingles under the broiler and cook the fish for about 5 to 7 minutes, until firm but not dry. Remove the fillets to a platter or serve directly from the shingles.
Just before serving, stir the parsley and grated eggs into the reserved warm egg sauce. Fold in chives and caviar if using. Top the fillets with the egg sauce and a sprinkling of chives, and a dollop of caviar if using.