Peel the prickly pears carefully, blend until smooth, and strain to a fine purée for the gazpacho base.
Whisk in lemon juice, olive oil, and sugar; season with sea salt and black pepper.
Fold in the diced cucumber, onion, and tomatoes. Chill well.
Ladle into chilled bowls, crown each with burrata, garnish with chopped olives and torn basil, and finish with a thread of EVOO.