Place the squash halves face-down in a baking dish in 1 inch of water and bake until flesh is tender, about 45 minutes. Meanwhile, heat the olive oil in a large skillet over medium heat; cook the sausage in the oil until brown, breaking the meat into small pieces as it cooks; set aside.
Once the squash halves are tender, discard the water. Generously sprinkle brown sugar into the cavity of each squash half, reserving about 1/4 cup for later use. Fill the squash halves with the browned sausage, and top with the reserved brown sugar. Return squash to the baking dish and bake until the sugar begins to melt, about 15 minutes.
