In a large skillet, melt butter over medium-high heat and let it brown, watching carefully so it doesn’t burn. As soon as the butter is brown and smells nutty, scrape in shallots, turn heat down to medium, and stir and sauté till they soften. Pour in mushrooms, add ½ teaspoon of salt, and stir shallots and mushrooms together. Cook mushrooms with shallots and browned butter until they are soft and browned, turning occasionally. This should take about 15 minutes. When the mushrooms are browned and most of their liquid has evaporated, pour in the sherry and stir in the thyme, scraping up the brown bits on the bottom of the pan. Continue to sauté the mushrooms until the liquid has reduced and thickened. Taste and add salt and fresh-ground black pepper as desired.
For the Clafoutis:
Heat oven to 375. In the bowl of a mixer fitted with a whisk attachment, blend the eggs with salt until frothy. Add milk and mix well. Add the flour and mix until frothy and thoroughly incorporated, 1-2 minutes, scraping the sides of the bowl as needed. Pour the batter over the mushrooms. Sprinkle the top with the grated Gruyere. Grind a little black pepper over the top. Place the dish in the center of the oven. Bake until the center puffs and turns a golden color, and the clafoutis is set, about 30-40 minutes. (Cover with foil if the top browns too quickly.) Serve warm or at room temperature.