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Using a small sharp knife and working with one chicken breast at a time, cut a 2-inch-long horizontal slit into the thick side of chicken breast. Move knife back and forth in slit to form a pocket. Divide cheese mixture among chicken pockets. Press edges closed and secure with a toothpick to seal. Sprinkle chicken with salt and pepper.
Heat oil in large ovenproof skillet over high heat. Add chicken, skin-side down; cook for 3 minutes or until skin is brown. Turn chicken over; transfer skillet to oven. Bake until cooked through (170 degrees F), about 10 minutes.
Meanwhile, in a heavy small saucepan, melt butter over medium heat. Remove from heat and stir in lemon juice. Slice chicken and place on a serving plate. Drizzle lemon butter over chicken. Garnish with additional basil.