Heat 1 tablespoon each olive oil and butter in a large, heavy, nonreactive sauté pan.
Season the chicken with salt and freshly ground black pepper and add to the pan when the oil and butter are hot and bubbly. Sear the 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 the remaining olive oil and butter to the pan and, when hot, stir in the shallots and cook until soft and lightly caramelized, about 4 minutes.
Add the blueberries and stir 1 minute, followed by the balsamic vinegar, maple syrup, rosemary, salt and pepper and simmer about 10 minutes, or until the blueberries have collapsed.
Pour the blueberry balsamic mixture over the chicken and put the pan in the top half of the oven for about 10 minutes, until the chicken has cooked through to a temperature of 170 degrees F.