Make the lemon-thyme oil: Combine the olive oil, garlic, lemon peel, and thyme in a small saucepan over low heat. Gently warm the mixture, then turn off the heat. Let steep for 20 minutes, then strain.
Make the burrata platter: Arrange the tomatoes, peach slices, and burrata on a platter. Sprinkle with salt, then drizzle with some of the lemon-thyme oil. Top with the basil and mint and sprinkle with the pistachios. Top with cherries and currants, if desired. Season to taste with salt and pepper and serve with toasted bread.