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Preheat oven to 350°F.
Heat oil in large skillet set over medium heat. Add onions; cook, stirring often, for 25 to 30 minutes or until very soft and caramelized. Add apples, garlic, sage, thyme, salt, pepper and wine. Cook, stirring, for 3 minutes or until apples are softened. Let cool slightly.
Lay chicken breasts on flat work surface. Starting at long side of breast and holding long carving knife parallel to cutting board, cut each breast in half without cutting all the way through; open up like a book.
Bake in center of preheated oven for 25 to 30 minutes or until juices run clear. Spoon pan juices over each breast, remove toothpicks and serve immediately.
Spread stuffing over half of each breast. Cover stuffing with the top half. Using toothpicks, secure open side so that stuffing does not fall out.
Transfer to foil-lined baking sheet. Sprinkle each breast with salt and pepper.