How to Make It
Preheat oven to 400°F. Heat 4 teaspoons oil in a medium saucepan over medium-high. Add mushrooms and shallots; cook, stirring often, until shallots are softened, about 3 minutes. Stir in wine, 2 tarragon sprigs, 2 thyme sprigs, and 2 fennel sprigs. Return mixture to a simmer over medium-high, and cook until wine is reduced by half, 4 to 5 minutes. Stir in stock and cream, and return to a simmer. Cook, stirring occasionally, until reduced to about 2/3 cup, 18 to 22 minutes.
Remove sauce from heat, and gradually whisk in 1/2 cup butter. (You may need to return saucepan to heat briefly to melt butter; do not let mixture come to a simmer or sauce will break.) Pour sauce through a fine wire-mesh strainer into a small heatproof bowl; discard solids. Stir in lemon juice, 1/2 teaspoon salt, and 1/4 teaspoon pepper; cover to keep warm.
Season tops of salmon fillets with remaining 1 1/4 teaspoons salt and remaining 1/2 teaspoon pepper. Heat 1 tablespoon oil in a large nonstick or cast-iron skillet over medium-high. Add 5 salmon fillets, flesh sides down, and sear until lightly browned, about 1 minute. Flip fillets, and sear until skin is lightly browned, about 1 minute. Transfer cooked fillets, skin sides down, to an aluminum foil–lined rimmed baking sheet, spacing fillets 1/2 inch apart. Wipe skillet clean, and repeat process with remaining 1 tablespoon oil and remaining 5 salmon fillets.
Top fillets with remaining 1/2 cup cubed butter, remaining 8 tarragon sprigs, remaining 8 thyme sprigs, and remaining 8 fennel sprigs. Bake in preheated oven to desired degree of doneness, 6 to 8 minutes for medium.
Spoon about 2 tablespoons warm wine sauce onto each of 10 dinner plates. Using a spatula, carefully slide salmon fillets off their skins, leaving skins and herb sprigs on baking sheet. Pat bottoms of cooked fish with paper towels. Place fillets on sauce, skinned sides down. Garnish with tarragon leaves and fennel fronds, and serve immediately.