Make the pistachio vinaigrette. Combine pistachios, basil, balsamic vinegar, lemon juice, water, dijon mustard, salt, pepper, and olive oil in a blender. Puree until smooth and set aside.
Toss the cut peaches in 1 tablespoon olive oil. Grill the peaches until grill marks appear and peaches begin to soften, about 4-5 minutes. Remove from heat and slice into wedges.
Add corn to the grill and cook until lightly charred. Remove from heat and shuck the corn.
Combine corn, peach wedges, halved tomatoes, and sliced shallots in a large bowl. Add pistachio vinaigrette and toss to combine.
Transfer salad to a platter and top with sliced fresh mozzarella. Season salad with salt and pepper and sprinkle pistachios, fresh basil, and mint. Drizzle extra vinaigrette on top and serve.
Notes
You can make the vinaigrette up to 3 days in advance. It will thicken as it sits, so you can add a splash of water to help loosen it up.Use fresh, ripe but firm peaches for the best results.