Into a food processor, add the cheese, butter, mustard, and garlic. Blend until the mixture is thick, smooth and spreadable. Spread the cheese mixture over 1 side of each bread slice. Arrange the turkey slices over the cheese mixture on 6 of the bread slices. Top sandwiches with the remaining bread slices, cheese mixture side down, pressing gently to adhere. Cut the sandwiches in half.
Heat 3 tablespoons of oil in a heavy skillet over medium-low heat. Cook half of the sandwiches until golden brown and heated through, about 4 minutes per side. Transfer to a serving platter and tent with foil to keep them warm. Repeat with remaining 3 tablespoons of oil and sandwiches. Serve immediately.