Preheat oven to 375°. In a large, ovenproof (not nonstick) frying pan over medium-high heat, sauté mushrooms, thyme, 1/2 tsp. salt, and the pepper in 2 1/2 tbsp. oil until mushrooms are crisp and brown, 8 to 10 minutes. Add wine and garlic and cook 1 minute more. Transfer to a bowl.
Wipe out pan; pour in 1 1/2 tbsp. oil. Salt fish lightly on skinned side and set it, skinned side down, in pan. Cook 5 minutes on stove, then bake until fish is almost opaque, 3 to 6 minutes.
Transfer fillets to plates. Pour out liquid from pan, scrape pan clean, and set over medium-low heat. Reheat mushrooms, pour off any excess liquid, and stir in sour cream. Spoon mushrooms over fish.