How to Make It
Preheat the oven to 400°. In a large roasting pan, season the capon inside and out with salt, pepper and the herbes de Provence. Roast the capon breast side up for about 30 minutes, then turn it breast side down and roast for about 1 hour. Turn the capon breast side up and roast for about 1 hour longer, until an instant-read thermometer inserted in the inner thigh registers 160°.
Meanwhile, in a large, deep skillet, melt the butter. Add the white and shiitake mushrooms and season lightly with salt and pepper. Cover and cook over moderate heat, stirring occasionally, until the mushrooms are lightly browned, about 15 minutes. Add the vermouth and cook until almost evaporated. Add the stock and simmer over moderate heat for 5 minutes. Add the cream and Armagnac and simmer until the sauce has reduced to 3 cups, about 4 minutes. Remove from the heat and whisk in the potato starch mixture until incorporated. Cook over low heat, whisking, until thickened, about 1 minute. Remove from the heat, stir in the tarragon and season with salt and pepper.
Lift the capon from the roasting pan, letting the cavity juices drain into the pan; transfer the capon to a carving board and let rest for 15 to 20 minutes.
Pour the pan juices into a glass measuring cup and skim the fat from the surface. Strain the juices into the mushroom sauce and reheat gently. Carve the capon at the table and serve with the mushroom sauce.