About 2 hours before serving or early in day: on cutting board, holding knife parallel to work surface, and starting from a long side of turkey breast, cut breast horizontally, almost, but not all the way through. Spread breast open to make a butterflied breast. With meat mallet or dull edge of French knife, pound butterflied breast between 2 sheets plastic wrap into a rectangle about 12 inch by 10 inch.
Chop enough basil to equal 1/4 cup; set aside. Arrange basil leaves to cover entire turkey breast; top basil with sliced ham, then sliced cheese. Starting from long side, roll turkey, jelly roll fashion, to enclose stuffing completely.
Tie turkey breast roll securely with string at 1 1/2 inch intervals; place on rack in small roasting pan.
In cup, mix olive or salad oil, dried basil leaves, pepper and 1/4 teaspoon salt; brush over turkey breast roll. Insert meat thermometer into center of roll. Roast turkey breast roll in 325 oven 1 1/4 to 1 1/2 hours, brushing occasionally with any pan drippings, until thermometer reaches 170.
Meanwhile, in medium bowl, mix mayonnaise or whipped salad dressing, vinegar, sugar, reserved chopped basil, and 1/2 teaspoon salt; refrigerate until ready to serve.
When turkey is done, remove to cutting board and discard string. Slice turkey roll crosswise into 1/2 inch thick slices. Arrange slices on platter. Garnish with remaining basil leaves. Serve warm, or, cover and refrigerate to serve cold later. Serve with basil sauce. Makes 10 main dish servings.
Each serving: about 340 calories, 25 g fat, 76 mg cholesterol, 510 mg sodium.
