Add all salad dressing ingredients to a mason jar and top with a lid. Shake until homogenous. Set aside.
Preheat grill or grill pan to medium high heat. Brush insides of peach halves with olive oil and lightly season with salt and pepper. Cook, insides down, on the grill for 3-4 minutes until beautiful char marks develop. Flip and cook for just about 1-2 more minutes. Remove from grill.
Add arugula to a mixing bowl with a few tablespoons of dressing. Toss to coat, adding more dressing if needed.
To serve, plate arugula and top with grilled peaches and burrata halves. Serve with extra dressing, if desired.