Preheat a grill pan or outdoor grill to medium-high heat.
Quarter the peaches, removing the flesh from the pits. Toss the peaches with 2 tablespoons oil and some salt and pepper in a large bowl to coat. Grill the peach pieces, turning once, until charred on both sides, about 2 minutes per side.
Meanwhile, whisk together the vinegar, chile paste and the remaining 2 tablespoons oil in the same bowl. Toss the grilled peaches in the dressing, then place on a serving platter and drizzle with the remaining dressing. Sprinkle with the mint and basil. Nestle the burrata into the peaches and score the top with an "x" to expose the creamy inside.