Whisk extra-virgin olive oil, white balsamic vinegar, honey, kosher salt, and black pepper in a small bowl until emulsified.
Place arugula in a large bowl, drizzle with 2 tablespoons of the dressing, and toss to lightly coat. Spread on a serving platter.
In the same bowl, gently toss the sliced peaches with 1 tablespoon of the dressing. Arrange the peaches over the arugula.
Tear the burrata over the salad and drizzle with the remaining dressing.
Toast the pistachios in a dry skillet over medium heat, stirring, until fragrant, 3–4 minutes; transfer to a board and roughly chop. Tear the basil leaves. Scatter the pistachios and basil over the salad and serve immediately.