Gently clean the mushrooms with a damp cloth or a paring knife to remove any dirt or dark spots. Cut off any tough stems. If the mushrooms appear muddy, quickly dip them into a large basin of water and drain. Leave small, bite-size mushrooms whole; quarter or halve larger mushrooms.
Melt 1 Tbs. of the butter together with 1 Tbs. of the oil in a 10-inch straight-sided sauté pan over medium-high heat. Add the mushrooms and a generous pinch of salt, and cook, stirring frequently, until any liquid has evaporated and the mushrooms are nicely browned, 5 to 8 minutes. (If the mushrooms are dry and the pan begins to scorch, add a drizzle of olive oil.) Remove the pan from the heat and transfer the mushrooms to a cutting board. Let them cool slightly and chop them coarsely.
Return the pan to the stovetop over medium heat and add the remaining 1 Tbs. butter and 1 Tbs. oil. When the butter has melted, add the shallots, thyme, and a pinch of salt. Cook, stirring, until the shallots are tender and lightly golden, about 3 minutes. Return the mushrooms to the pan, stir in the crème fraîche (if you’re making this ahead, see the note below), and cook, stirring, to coat the mushrooms with the crème fraîche. Stir in the parsley and season with several grinds of pepper. Season with more salt and pepper to taste. Remove from the heat and hold in a warm spot.
Shortly before serving, position an oven rack about 6 inches from the broiler element and heat the broiler to high. Arrange the bread slices on a baking sheet and brush them with olive oil. Broil until the bread is golden, 1 to 2 minutes. Flip and toast the other side, about 1 minute. Spread the warm mushroom mixture on the toasts, sprinkle some of the Parmigiano-Reggiano on top, and serve.
The mushroom topping can be made several hours ahead and refrigerated, but hold back about half of the crème fraîche. When you’re ready to serve, reheat the mushrooms over low heat and add the rest of the crème fraîche (don’t overheat or the cream will break).