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PREHEAT oven to 350°F.
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PREHEAT a large, heavy, nonreactive skillet.
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SEASON chicken with 1/2 teaspoon salt.
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SEASON chicken with 1/4 teaspoon pepper.
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ADD 1 tablespoon olive oil, 1 tablespoon butter, and chicken to pan when oil and butter are hot and bubbly.
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SEAR chicken over medium-high heat about 1 minute per side or until golden brown and a light crust forms.
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TRANSFER chicken to a baking dish and SET ASIDE.
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REHEAT skillet.
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ADD remaining olive oil and remaining butter.
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STIR IN 2 shallots.
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COOK shallots about 4 minutes or until soft and lightly caramelized.
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ADD blueberries and COOK 1 minute.
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ADD 1/2 cup vinegar, 1/3 cup maple syrup, 1 tablespoon rosemary, remaining salt, and remaining pepper.
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STIR to combine ingredients.
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SIMMER about 10 minutes or until blueberries have collapsed.
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POUR blueberry mixture over chicken.
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PLACE baking dish into oven on top shelf.
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BAKE about 10 minutes or until chicken has cooked through to a temperature of 165°F.
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SERVE immediately.