To make the pesto: Add the basil leaves, garlic, pine nuts, and salt to the bowl of your food processor fitted with the blade attachment. Pulse 4-5 times until coarsely chopped. Then with your food processor running, slowly drizzle in the olive oil and process until well combined. Transfer the pesto to a clean container. Cover and refrigerate it until you're ready to use it. The pesto can be made 1-3 days in advance.
To make the hot honey: Stir together the honey and hot sauce until well blended. Set aside.
Pre-heat your grill to medium-high.
Add the peach wedges to a large baking dish and drizzle them with about 2 tablespoons of the olive oil. Using a pastry brush and/or your hands, brush/rub the olive oil over all sides of the peaches so they are well coated with a thin layer of oil. Use the remaining olive oil to coat both sides of the bread slices.
Place the peach wedges and bread slices on the grill. Cook the peach wedges about 1-2 minutes per side, or just until they develop a few grill marks and start to soften slightly. Cook the bread slices just until they are toasted and crisp. Remove everything from the grill.
Assemble the appetizer by arranging the grilled peach wedges down the center of a serving plate/tray. Drizzle with some of the hot honey and a couple tablespoons of the basil pesto (you can stir in more olive oil if needed to make it thin enough to drizzle easily). Arrange the bread slices and burrata cheese around the grilled peach wedges. Season the tops of the burrata cheese with a pinch of sea salt and freshly ground black pepper.
Serve immediately with the remaining hot honey and basil pesto on the side.