The flesh immediately surrounding the seeds is the most succulent part of any melon; if you haven’t tried eating the crunchy seeds before, hold on to your hat.
Combine vinegar, kosher salt, 1/4 teaspoon pepper, and 2 tablespoons water in a large bowl; add melon and toss to coat. Let stand at room temperature at least 30 minutes.
Remove melon from pickling liquid and arrange on a platter with tomato and burrata. Drizzle with oil and some pickling liquid, top with basil, and season with sea salt and pepper.
Do ahead: Melon can be pickled 2 hours ahead. Cover and chill.
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