Heat oven to 350ºF.
Melt the clarified butter In a 12-inch heavy sauté pan over high heat. When the butter begins to shimmer, add the sliced mushrooms and cook, stirring frequently, until all the liquid has released and mushrooms are golden brown, 12 to 15 minutes.
Make a space in the middle of the pan and add the shallots. Cook for about 30 seconds then stir to combine with the mushrooms. Kill the heat and add the cognac, scraping up any browned mushroom bits. Season with 1 teaspoon of the salt, the black pepper, and the chives.
Return the pan to medium heat and add the cream, Parmesan, and tarragon. Stir to combine and let simmer for 1 minute. Remove the pan from the heat and add 1 tablespoon of the breadcrumbs. Allow the filling to cool while preparing the caps.
Place the mushroom caps in a large bowl and toss with the olive oil to evenly coat. Add the rosemary, thyme, garlic, and the remaining salt and toss to combine thoroughly.
Place the mushroom caps concave side (where the stem used to be) down on a half-sheet pan fitted with a wire rack. Roast the mushroom caps until the tip of a paring knife can be inserted into the side of the mushroom with little or no resistance, about 10 minutes.
Remove the mushroom caps from the oven and crank the setting to high broil. Flip the caps over and fill each cavity with about 1 tablespoon of the cooked mushroom filling. Top each cap with the bread crumbs and broil until the tops have browned, 3 to 4 minutes. Serve immediately.