Preheat oven to 450 degrees F. Place a "V" rack into a large roasting pan. Pour 2 cups of water into the bottom of the roasting pan and set aside.
In a small bowl, combine unsalted butter, chopped fresh herbs, and 1 tablespoon lemon juice. Set aside.
Remove neck and giblets from the capon's cavity and discard. Season the capon on all sides with salt and pepper. Place garlic cloves, juiced lemon halves, and whole fresh herbs into the capon's cavity. Tie legs together with butcher's twine. Rub the entire exterior of the capon with the herb butter.
Place the capon, breast-side down, onto the prepared rack. Roast for 20 minutes. Remove from oven and reduce temperature to 375 degrees F. Using two meat forks or poultry lifters, carefully flip the capon so that the breast is facing up. Baste the capon with pan drippings.
Return capon to the oven and roast for 60 minutes. Check the pan drippings; if the water has evaporated, add another 1-2 cups to the bottom of the roasting pan.
Continue roasting another 60 minutes (approximately), basting occasionally, until a probe inserted into the meatiest parts of the breast and thigh (away from the bone) registers 165 degrees F and the juices run clear.
Transfer the capon to a carving board and tent with foil. Let rest for 15 minutes. Carve and serve with Mushroom-Cognac Sauce.
Make the Mushroom-Cognac Sauce
Clean mushrooms and slice 1/8- to 1/4-inch thick, discarding any tough stems.
In a large sauté pan, melt butter until foaming subsides. Add mushrooms, a pinch of salt, and a few grinds of freshly-ground pepper, and cook over medium-high heat until liquid evaporates and mushrooms are lightly-browned.
Stir in vermouth and cook until almost entirely evaporated, 1-2 minutes. Add chicken stock, reduce heat to maintain a steady simmer, and cook for 5 minutes. Remove from heat and stir in heavy cream and cognac.
Return to stove and simmer until sauce is slightly reduced and thickened, an additional 5 minutes. Whisk in dissolved cornstarch until smooth. Cook over low heat until sauce is velvety, about 1-2 minutes more.
While the capon is resting, loosen any browned bits from the bottom of the roasting pan with a wooden spoon or whisk. Strain pan drippings into a gravy separator or liquid measuring cup, allowing the fat to separate.
Discard fat and whisk about 3 tablespoons of pan drippings into the mushroom sauce. Stir in chopped tarragon and season to taste with salt and pepper.
If sauce is too thick, thin with a little bit of water or chicken stock until desired consistency is reached. Serve sauce alongside carved capon.
*You can use a variety of mushrooms in this recipe. Jacques Pépin's original recipe calls for shiitake, but the cremini looked freshest at my local market the day I went shopping, so I opted to use them instead. A gourmet mushroom blend would also be lovely here.