Heat oven to 350ºF.
In a 12-inch heavy sauté pan, heat the clarified butter over high heat. When the butter begins to shimmer, add the sliced mushrooms one handful at a time to the sauté pan. Turn the mushrooms over as they begin to color. Continue to cook until brown and tender, 12 to 15 minutes.
Make a space in the middle of the pan and add the shallots, stir to combine. Deglaze the pan with cognac, scraping up any browned mushroom bits. Remove the pan from the heat and add the chives. Adjust seasoning as desired.
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.
In a large bowl, toss the mushroom caps with enough olive oil to coat the caps. Add the rosemary, thyme, and garlic and toss to combine thoroughly.
Place a roasting rack on a baking sheet. On the baking sheet, turn the mushroom caps upside down, stem-side up. 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.
Heat broiler to high and move the oven rack up one level. Mound 1 tablespoon of filling into each mushroom cap; avoid overstuffing the caps. Top each cap with enough bread crumbs to cover the filling. Broil the mushroom caps on high until the filling bubbles and the tops have browned, 3 to 4 minutes.