To prepare the vinaigrette, combine the ingredients in a jar with a tight-fitting lid and shake vigorously until emulsified.
To prepare the salad, brush the cut sides of the peaches with olive oil and season lightly with salt.
Grill the peaches, cut-side down, over medium-hot coals or in a cast-iron grill pan over high heat.
When grill marks appear, turn over the peach halves and grill briefly on the skin side—about 1 to 3 minutes.
Cool the peaches slightly, then cut each half into two pieces.
Arrange on a serving platter and tuck basil leaves, prosciutto and burrata pieces around.
Drizzle with the vinaigrette and season to taste with salt and freshly ground pepper.