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Heat butter and oil in a large skillet over medium-high heat. Season both sides of the chicken strips with salt and pepper. Add the chicken to the hot butter and oil; cook about 2 minutes on each side, or until chicken is golden brown on each side and cooked through. Remove to a platter; keep warm.
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Add wine to the hot skillet, stir to scrape up the browned bits on the bottom of the pan. Allow the wine to boil until it's reduced by about half. Add chicken broth and mustard to the skillet and whisk to combine.
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In a small bowl, whisk cornstarch and water; stir into skillet. Add dried cranberries. Boil 1 to 2 minutes or until sauce thickens.
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To serve, spoon sauce over the cooked chicken and top with green onions.