How to Make It
Preheat the oven to 350°. In a large ovenproof skillet, heat the oil until shimmering. Add the salmon skin side down and season with salt and pepper. Cook over high heat until the skin is lightly browned, about 3 minutes. Transfer the skillet to the oven and roast the salmon for about 10 minutes, until it is cooked through and the skin is very crisp.
Meanwhile, in a large skillet, melt 2 tablespoons of the butter. Add the mushrooms and cook over high heat, stirring occasionally, until lightly browned, about 6 minutes. Add the shallot, garlic and thyme and season with salt and pepper. Cook, stirring, until the shallot is softened, about 3 minutes. Add the wine and broth and simmer until the liquid is nearly evaporated, about 5 minutes. Remove from the heat. Stir in the remaining 2 tablespoons of butter.
Transfer the salmon to plates, skin side up. Serve with the mushrooms.