Preheat your barbecue until the flames have subsided and the coals have turned white.
Grill the peppers in their skins until completely blackened all over, then transfer to a bowl and cover with clingfilm (plastic wrap) to steam for a few minutes. When cool enough to handle, peel off the blackened skins, slice the peppers into thick slivers and set aside.
In a bowl, toss the nectarine slices in 1½ tbsp of the olive oil, until well coated. Grill the slices until they are a little charred, then set aside.
Using your fingers, sprinkle the bread slices with 2 tbsp of the red wine vinegar, then toast on the grill and set aside.
In a large bowl, mix the red onion with the sugar and lemon juice and leave to macerate for around 20 minutes.
Add the remaining 1 tsp of red wine vinegar, the remaining 3 tbsp of olive oil and the garlic to the onions, then add the sliced, cooked peppers, the tomatoes with their juices, the grilled nectarine slices and the basil. Tear the toasted bread into rough chunks and add to the mix, then toss to combine. Season to taste.
Transfer the salad to a large platter then tear over the burrata, drizzling with a little more olive oil if you like.