1 ½ pounds ripe but slightly firm peaches (4 peaches), halved and pitted
2 tablespoons unsalted butter, melted
12 ounces ripe tomatoes, cored and cut into ½-inch pieces
¾ teaspoon table salt, divided
5 tablespoons extra-virgin olive oil, divided
1 tablespoon white wine vinegar
8 ounces burrata cheese, room temperature
⅓ cup chopped fresh basil
FOR THE PEACHES: Brush cut side of peaches with melted butter.
FOR A CHARCOAL GRILL: Open bottom vent completely. Light large chimney starter three-quarters filled with charcoal briquettes (4½ quarts). When top coals are partially covered with ash, pour evenly over half of grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.
FOR A GAS GRILL: Turn all burners to high; cover; and heat grill until hot, about 15 minutes. Leave primary burner on high and turn off other burner(s).
Clean and oil cooking grate. Arrange peaches cut side down on hotter side of grill and cook (covered if using gas) until grill marks have formed, 5 to 7 minutes, moving peaches as needed to ensure even cooking.
Transfer peaches cut side up to 13 by 9-inch baking pan and cover loosely with aluminum foil. Place pan on cooler side of grill. If using gas, turn primary burner to medium. Cover and cook until peaches are very tender and paring knife slips in and out with little resistance, 10 to 15 minutes. When peaches are cool enough to handle, discard skins, then let cool completely.
FOR THE SALAD: While peaches cool, toss tomatoes with ¼ teaspoon salt and let drain in colander for 30 minutes. Cut each peach half into 4 wedges and cut each wedge in half crosswise.
Whisk vinegar and remaining ½ teaspoon salt together in large bowl. Whisking constantly, slowly drizzle in ¼ cup oil. Add peaches and tomatoes and toss gently to combine; transfer to shallow serving bowl. Place burrata on top of salad and drizzle with remaining 1 tablespoon oil. Season with pepper to taste and sprinkle with basil. Serve, breaking up burrata with spoon and allowing creamy liquid to meld with dressing.