Heat 2 Tbs. of the oil in a 12-inch nonstick skillet over medium-high heat. Add the shallots and cook, stirring often, until golden brown, about 2 minutes. Add the mushrooms and cook, stirring occasionally, until softened, 3 to 4 minutes. Stir in the cream, thyme, 1 tsp. salt, and 1/8 tsp. pepper. Reduce the heat to medium and cook, stirring often, until thickened, 1 to 2 minutes. Remove the skillet from the heat and stir in the spinach until wilted. Transfer to a bowl and cover with foil to keep warm. Clean the skillet and return it to the stove.
Heat the remaining 1-1/2 Tbs. oil in the skillet over medium heat. Season the salmon fillets all over with 1 tsp. salt and 1/4 tsp. pepper. Arrange the salmon in the skillet skin side up in a single layer and cook, turning once, until golden brown and just cooked through, 7 to 9 minutes total. Transfer the salmon to a large plate.
Mix the lemon juice with 1 Tbs. water, add to the skillet, and cook, scraping up any browned bits, until just thickened, about 30 seconds. Drizzle the pan juices over the salmon and serve with the warm shiitake mixture.