Step 1Combine the squash, Brussels sprouts, brown sugar, butter, 1 tablespoon chopped sage, 1 teaspoon salt and one-half teaspoon pepper in a 9-by-12-inch glass baking dish and roast uncovered in a 375-degree oven, stirring several times, until the squash is tender, about 40 minutes.
Step 2Meanwhile, heat the grill. Rub the olive oil onto both sides of each chicken half. Season the chicken with one-half teaspoon salt and the remaining pepper. Place the chicken on the hot grill, skin side down, and cook for 15 minutes, then turn and continue cooking until the juices from the thigh run clear, about 35 to 40 minutes total.
Step 3Combine the hazelnuts, honey, remaining salt and remaining sage in a small skillet and cook over low heat to melt the honey and coat the hazelnuts, about 5 minutes.
Step 4To serve, spoon half of the roasted vegetables onto each dinner plate and nestle a chicken half nearby. Garnish with the honeyed hazelnuts.