Preheat grill pan or outdoor grill to medium-high heat. Brush peach halves with 1 tablespoon olive oil and season lightly with salt.
Grill peaches cut-side down for 3-4 minutes until caramelized with nice grill marks. Flip and grill another 2-3 minutes. Remove and drizzle with honey.
Toast bread slices until golden brown and crispy. Brush with remaining olive oil while warm.
Slice grilled peaches into wedges. Tear burrata into bite-sized pieces.
Arrange peach slices on toasted bread, top with burrata pieces, and drizzle with balsamic vinegar.
Garnish with torn basil leaves, season with freshly cracked black pepper, and serve immediately.
Notes
Storage tip: Grilled peaches can be made up to 2 days ahead and stored covered in refrigerator. Assemble toast just before serving for best texture. Nectarines, plums, or pears work well as peach substitutes.