Heat the oil in a 4-quart saucepan over medium heat. Add the shallot and 1/2 tsp. salt and cook, stirring, until beginning to brown at the edges, 3 to 4 minutes. Add the white and shiitake mushrooms, the thyme, 1/2 tsp. salt, and 1/4 tsp. pepper. Increase the heat to medium high, and cook, stirring, until the mushrooms soften and start to brown, 3 to 5 minutes. Add the sherry and cook, stirring, until reduced by half, about 2 minutes. Add the broth and 1 cup water; bring to a boil. Reduce the heat and simmer for 5 minutes. Stir in the Parmigiano and let cool briefly.
Working in batches, purée the soup in a blender until smooth. Transfer to a 3-quart saucepan and reheat.
In a small bowl, mix the crème fraîche, chives, and 1/8 tsp. black pepper.
Season the soup to taste with salt and pepper and serve topped with a swirl of the crème fraîche.