Put the grapes in a medium bowl with vinegar, oil, garlic, sugar, 1 teaspoon of the fennel seeds, 1/4 teaspoon of flaked salt, and plenty of pepper. Mix well and marinate for at least 1 hour and up to 1 day. Thread 5 or 6 grapes onto each skewer. Don't throw away the marinade; you'll need it when serving.
Place a grill pan over high heat and ventilate your kitchen well. Once hot, add the grape skewers in batches and grill for 2 to 3 minutes, turning after 1 1/2 minutes. Remove from the heat.
When ready to serve, tear the balls of burrata in half and place one-half on each plate. Arrange the grape skewers to lean against them—2 per portion—and spoon 1 1/2 teaspoon of the marinade over the cheese. Alternatively, arrange on a platter to serve a crowd. Sprinkle with the remaining 1/2 teaspoon of fennel seeds, garnish with a sprig of basil, and serve.