Using melon baller or small soup spoon, scoop out the insides of cucumber pieces from one cut side, leaving ¼ inch border around walls and bottoms, to form cups. Arrange cups on a serving platter, cover with damp paper towels, and refrigerate until ready to serve. Note: Cups can be wrapped in plastic and refrigerated for up to 24 hours.
In a blender, add 4 cucumber ends, ½ cup tomato juice, 1 cored and chopped tomato, ½ chopped bell pepper, 1 chopped shallot, 2 tablespoons oil, 1½ tablespoons vinegar, 1 clove minced garlic, ½ teaspoon salt, ¼ teaspoon pepper, and ½ teaspoon optional hot sauce. Process until mostly smooth, about 1 minute.
Transfer to 8-cup liquid measuring cup or bowl, cover, and refrigerate until completely chilled, at least 1 hour or up to 2 days.
Stir gazpacho to recombine and season with extra vinegar, salt, and pepper to taste.
Divide gazpacho among cucumber cups, drizzle with extra olive oil, and sprinkle with pepper.