Heat a grill or grill pan to medium-high. Clean and oil the grates so the fruit won’t stick.
Make the glaze: In a small saucepan, combine the balsamic vinegar and honey. Bring to a gentle boil, then simmer over medium-low heat until reduced by about half and syrupy, 8–10 minutes. Remove from heat to cool slightly.
Halve and pit the peaches. Brush the cut sides with 1 tbsp of the olive oil. Season the peaches with 0.25 tsp of the kosher salt and 0.25 tsp of the black pepper.
Grill the peaches cut-side down until marked and just tender, 3–4 minutes. Flip and grill skin-side 1–2 minutes more. Transfer to a board and slice into wedges, or leave as halves.
In a large bowl, combine the arugula with the lemon juice, 1.5 tbsp of the olive oil, and the remaining 0.5 tsp kosher salt and 0.25 tsp black pepper. Toss to coat, then arrange on a platter.
Nestle the warm grilled peaches over the arugula. Tear the burrata into large pieces and distribute over the salad.
Spoon the balsamic glaze over everything. Scatter the torn basil leaves on top and drizzle the remaining 0.5 tbsp olive oil. Serve immediately while the peaches are warm and the burrata is cool and creamy.