How to Make It
Preheat the broiler. In a small skillet, melt the butter in the olive oil over moderate heat. Add half of the shallots and garlic and cook, stirring often, until softened and fragrant, 2 to 3 minutes. Season with a pinch of salt and pepper. Scrape the mixture into a small bowl and let cool.
Loosen the skin from the chicken breasts without detaching it completely. Place a teaspoon of the cooked shallot and garlic mixture under the skin of each breast half and spread it over the meat. Top the mixture with a folded piece of prosciutto and 3 basil leaves. Smooth the chicken skin over the fillings and season lightly with salt and pepper.
Set the chicken in a medium roasting pan, skin side up. Broil for 5 minutes. Turn and broil for 5 minutes more, then turn again and broil for about 5 minutes longer, or until nicely crisped on top; take care not to burn the skin. Turn the oven temperature to 350° and bake the chicken for about 20 minutes, or until the juices run clear when the meat is pierced. Transfer the chicken to a plate and keep warm.
Pour the juices from the roasting pan into a small bowl; spoon off the fat and reserve the brown drippings. Place the roasting pan over moderate heat. Add the remaining chopped shallot and garlic and stir until softened and golden, about 2 minutes. Pour in the wine and bring to a boil. When the wine is nearly evaporated, pour in the chicken stock and heavy cream. Bring to a boil and simmer, stirring occasionally, until reduced to 1/2 cup, about 4 minutes. Stir in the reserved drippings and the minced basil and season with salt and pepper.
Place a stuffed breast half on each of 4 dinner plates. Spoon the sauce over or around the chicken and serve.