The sweetness of blueberries and maple syrup blend with the acidity of balsamic vinegar and fragrant rosemary to make a rich sauce for baked chicken. Searing the chicken breasts in the skillet before making the sauce gives it extra flavor.
Preheat oven to 350°F. Heat 1 tablespoon each oil and butter in a large, heavy, nonreactive skillet. Season chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper and add to the pan when the oil and butter are hot and bubbly.
Sear chicken breasts over medium-high heat until golden brown, about 1 minute per side or until a light crust forms. Remove to a baking dish and set aside.
Add remaining olive oil and butter to the skillet and, when hot, stir in shallots and cook until soft and lightly caramelized, about 4 minutes. Add blueberries and cook 1 minute. Stir in vinegar, maple syrup, rosemary, remaining salt and pepper and simmer about 10 minutes, or until the blueberries have collapsed.
Pour the blueberry balsamic mixture over chicken and put baking dish in the top half of the oven for about 10 minutes, until the chicken has cooked through to a temperature of 165°F.