Brush olive oil all over the chicken thighs. Liberally sprinkle with salt and pepper.
Melt butter in a large skillet over medium-high heat. Place the chicken thighs, skin-side down, in the butter. Sear until the skin is browned and crisp, about 5 minutes. Use a pair of tongs to flip the chicken thighs over and cook for another minute. Remove from the pan and place on a plate or platter. Set aside.
Remove most of the fat and butter that is in the skillet (leave about 2 tbsp).
Over medium heat, add the shallots and thyme and cook until softened, about 2 to 3 minutes. Add the garlic and cook, stirring often, for another 30 seconds.
Add the mushrooms and cook, stirring often, until they are softened and beginning to release some liquid, about 5 to 8 minutes (If the mushrooms are too dry during this step, add in another 1 to 2 tbsp of butter). Sprinkle with salt and pepper.
Add the flour to the mushrooms and stir to coat for about 1 minute.
Carefully whisk in the wine and bring to a boil. Lower the heat and whisk in the chicken stock, 1 tsp salt, and ½ tsp black pepper. Whisk and/or stir with a spoon to remove any clumps. Simmer for about 5 minutes until starting to thicken.
Nestle the chicken thighs in the sauce, skin-side up. Add in any collected juices, too. Turn the heat down a little (to medium-low). Cover the skillet and let simmer for 30 minutes.
Remove the chicken from the sauce and stir in the crème fraîche. Bring to a low simmer. Add the chicken back into the sauce. Serve at once with sauce spooned over the chicken either on a platter or on individual plates, passing additional sauce at the table.