Preheat a grill or grill pan on high.
In a large bowl, whisk together the red wine vinegar, honey, salt and pepper. Add the red onions and mix to combine. Set aside.
Toss the peaches in a medium bowl with a little bit of olive oil, salt, and pepper. Use a grill brush to make sure the grill grates are very clean, then brush them with a little bit of oil to prevent sticking. Grill the peaches, with the lid open, until nicely charred on both sides but still firm in the middle. Let cool, then cut each piece in half lengthwise or into a bit sized piece.
Whisk ¼ cup olive oil into the bowl with red wine vinegar and onions, then add the peaches.
Brush the bread liberally with olive oil, then grill until nice and charred. As soon as you remove it from the grill, rub the hot bread all over with the cut end of the garlic to release its oils. Slice bread into a large dice and add to the bowl along with the cucumbers.
Toss everything together until the bread absorbs most of the liquid, then sit for about 15 minutes to let the flavors meld.
Gently tear the basil leaves into the salad and toss to combine. Taste for seasoning and adjust as needed.
Transfer panzanella to a large platter, then break apart the burrata and scatter it across the top. Drizzle with a little more olive oil and season with a few cracks of black pepper. Garnish with more basil and serve.