Pat thighs dry and season with 1/2 teaspoon salt and pepper. Heat oil in a large skillet over medium-high heat and cook chicken for 6 to 7 minutes per side until browned and at least 165°F; transfer to a plate.
Add butter and mushrooms to the skillet. Cook for 8 to 10 minutes, stirring occasionally, until the released water evaporates and the edges brown.
Stir in shallot, remaining salt and thyme for 2 minutes. Sprinkle flour over the mushrooms and stir for 1 minute.
Add sherry and broth gradually, scraping the skillet, then simmer for 3 to 4 minutes until the gravy coats a spoon.
Return chicken and its juices and warm for 2 minutes over low heat. Serve with mushrooms and gravy spooned around the thighs.