Pop the stems from the mushrooms and reserve. Using a melon baller or spoon, gently scoop out any remaining stem from inside the mushroom caps. Set the caps aside. Finely chop the stems and bits from the caps (there should be about 1 1/4 cups).

Melt the butter with 1 tablespoon of the olive oil in a medium skillet over medium-high heat. Add the shallots and cook, stirring frequently, until softened, about 2 minutes. Add the chopped mushrooms and 1/4 teaspoon of salt, and cook, stirring occasionally, until all the liquid from the mushrooms has evaporated, about 3 minutes. Add the Madeira and thyme and cook until dry. Transfer the mushroom mixture to a medium bowl and cool slightly.

Add the breadcrumbs, Parmesan, creme fraiche, and parsley to the mushroom mixture and stir to combine thoroughly. Season with salt and pepper, to taste.

Preheat the oven to 375 degrees F. Lightly oil the bottom and sides of a skillet that is just large enough for the mushroom caps to fit snugly inside in a single layer. Pour the water into the skillet. Brush the outside of the caps with 1 tablespoon of olive oil. Set the caps, smooth side-down, in the skillet. Divide the filling evenly among the caps. Drizzle the tops with the remaining 1 tablespoon of olive oil. Bake until the mushroom caps are soft and cooked and the filling is browned, about 35 minutes. Remove from the oven and serve hot or at room temperature.

Cook's Note: If desired, reduce the pan juices and drizzle over the stuffed mushrooms before serving.