Preheat a grill or grill pan over medium-high heat. Add some olive oil and grill peaches cut-side down for 3–4 minutes, until grill marks appear, and peaches soften. Remove from heat and set aside.
In a food processor, combine basil leaves, pistachios, garlic, lemon zest, and a pinch of salt. Pulse to break down the ingredients. Slowly add the olive oil and lemon juice while blending.
Gradually add cold water, a bit at a time, until the pesto reaches your desired smooth and slightly loose consistency. Stir in Parmesan cheese and check seasoning, adding salt and pepper as needed. Set aside.
Lay arugula on a serving plate. Top with grilled peach slices and halved cherry tomatoes. Tear the burrata over the top. Generously drizzle with the lemon-zesty pistachio pesto. Season with salt and pepper
