To make potatoes: Preheat oven to 375 degrees. Place a roasting pan in the oven for 5 minutes to preheat. In a mixing bowl, combine the potatoes, garlic, and olive oil and stir to coat the potatoes and garlic. Season generously with salt and pepper. Transfer to the roasting pan and roast, stirring occasionally until potatoes are soft when pierced with the tip of a knife and the garlic cloves are very soft and sweet in flavor, about 40 minutes. Remove from oven and keep warm.

To make mushrooms: In a 12-inch saute pan, heat the butter over medium heat. Add the shallots and cook for about 3 minutes, until softened. Add the garlic and cook about 1 minute longer. Add the mushrooms and thyme. Cover the pan and cook for 7 to 10 minutes, until the mushrooms are tender and have released their juices. If the mushrooms seem dry during cooking, add about 1/4 cup of the chicken stock.

When the mushrooms are soft, add the rest of the chicken stock, raise the heat to high, and cook until the liquid is reduced by half, about 6 minutes. There will be quite a bit of liquid in the mushrooms. Remove them from the heat, cover, and set side to keep warm.

To make the chicken:

Heat the oil and 1 tablespoon of the butter in a 12-inch saute pan over medium-high heat. Season the chicken breasts with salt and pepper. Place the chicken breasts in the pan, skin side down. Cook for about 6 minutes, until golden brown. Turn and cook until the juices run clear when pierced with the tip of a knife, about 6 more minutes. Add the rosemary and cook, basting frequently, for 6 to 8 minutes longer, until the chicken is cooked through and nicely browned. Transfer to a serving platter and set it aside, covered, to keep warm.

Pour off any oil in the pan and add the wine and Cognac. Raise the heat to high and deglaze the pan by scraping up any browned bits with a wooden spoon. Add the stock and cook until the sauce is reduced nearly by half or thickened and richly flavored, about 4 minutes. Remove the pan from the heat and swirl in the remaining 3 tablespoons of butter. Season with the chives, salt, and pepper. Spoon the sauce over the chicken and serve, passing the mushrooms and potatoes on the side.