In a medium bowl, combine the corn, peaches, chile, basil, white balsamic, extra-virgin olive oil and kosher salt. Mix well with a spoon. Let marinate for 30 minutes at room temperature or up to 2 hours in the refrigerator.
Meanwhile, heat a grill or grill pan over medium-high heat. Drizzle the ciabatta slices with the olive oil and grill until golden brown, about 1 minute per side.
To serve, tear off small amounts of burrata and place on the toast. Sprinkle with the flaky sea salt. Spoon some of the salad on top. Drizzle with a touch more olive oil if desired.