Preheat the oven to 360 degrees F.

Season the guinea hen on both sides with the salt and pepper. Set a large, oven-safe 14-inch saute pan over medium-high heat, and add the olive oil. Once the oil is hot, place the hen pieces in the pan, skin side down, and sear for 3 minutes on each side. Repeat with any remaining pieces and set aside on a platter or baking sheet. Add the onions and carrots to the pan and saute until lightly caramelized, about 2 minutes. Add the garlic and saute for 30 seconds. Deglaze the pan with the balsamic vinegar, scraping any browned bits on the bottom of the pan. Reduce the balsamic to about 3 or 4 tablespoons, about 2 to 3 minutes. Add the stock to the pan and return the seared guinea hen pieces to the pan, skin side up. Place the potatoes, thyme and sage in the pan and cover with a lid. Place the pan in the oven and cook until the guinea hens are tender, about 1 1/2 hours. Remove the pan from the oven and serve the guinea hens with the potatoes, carrots and the sauce spooned over top.