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Coat chicken with remaining 2 teaspoons oil; season lightly with salt and pepper.
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Grill or broil 4 to 5 minutes per side, until internal temperature is 170°F.
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Serve chicken surrounded by remaining sliced berries and warm strawberry sauce.
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Hull and slice strawberries. Set aside.
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Heat 2 teaspoons oil in a small saucepan over medium-high heat.
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Add shallots, garlic, bay leaf and thyme and stir until lightly browned, 2 to 3 minutes.
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Add 3/4 cup sliced strawberries and cook for 1 minute, stirring constantly. Add balsamic vinegar and simmer 1 minute.
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Add wine and simmer until liquid is reduced to about half.
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Add broth and simmer 5 minutes. Whisk cornstarch and water together. Add mixture to saucepan and boil for 1 minute, stirring until thickened. Strain sauce and season with salt and pepper. Reheat just before serving.