Whisk together the vinegar, lemon juice, mustard, honey, shallot, kosher salt, and pepper in a small bowl.
While whisking, slowly drizzle in the olive oil and whisk until the dressing is emulsified.
Arrange the mozzarella, peaches, tomatoes, and basil on serving platter or individual plates.
Drizzle with the dressing; sprinkle with flaky salt, and serve.