- Slice tomatoes in half. Cut chili into thin rings and shallot into thin strips.
- In separate bowl, combine remaining ingredients except for basil leaves. Add chili and shallot. Toss tomatoes with dressing and allow to marinate for at least one hour. Reserve basil leaves for assembly.
- In small pot, combine all ingredients except cucumbers. Heat briefly to dissolve sugar. Remove from heat and let cool to room temperature.
- Add cucumbers to lime mixture and purée in blender. Place in refrigerator to cool.
- When cool, spread granita into thin layer in bottom of pan. Place in freezer.
- Every 45 minutes for 3 hours, scrape granita using fork. Granita is finished when totally frozen but still light and fluffy.
- Gently toss about ¾ basil leaves with tomato mixture, being careful not to bruise tomatoes. Place into cold bowl and garnish with the remaining leaves. Sprinkle about 2 heaping tablespoons of granita on top.
- Serve immediately.