Spread each tortilla lightly with cream cheese. Arrange two ham slices across the middle of each tortilla. Add a layer of fresh basil, then a layer of tomatoes. Do not place ingredients too close to tortilla edges. Add lettuce; use enough leaf lettuce to cover across the center of each tortilla, with some of the frilly leaf edge hanging over.
Starting at one end, tightly roll up each tortilla. Stick toothpicks in four evenly spaced spots of tortilla roll. Place tortilla rolls in a dish, cover, and refrigerate for 2 hours to chill.
Slice each roll into four evenly sized sandwiches, and serve as soon as possible.