Toss the mozzarella with the basil and a pinch of salt in a small bowl. Whisk the vinegar with 3 tablespoons of the olive oil in a separate small bowl until emulsified. Set both aside.
Brush the tomatoes and corn with the remaining 3 tablespoons oil and season with salt and pepper. Grill the corn, rotating occasionally, until cooked and charred in spots, about 20 minutes. Grill the tomatoes, rotating occasionally, until the skins start to burst and are charred in spots, about 7 minutes.
Peel the skins away from the tomatoes, then chop the tomatoes into small cubes. Transfer to a small bowl and season with salt and pepper. Slice the kernels from the cobs, transfer to a small bowl and season with salt and pepper.
Spoon 2 tablespoons corn into the bottom of each of 6 half-pint mason jars. Top each with 2 tablespoons tomato, then 2 tablespoons mozzarella. Repeat the process one more time so there are 6 layers total, ending with the mozzarella on top. Drizzle with some of the cider vinaigrette. The salads will keep, refrigerated, for up to 2 hours.
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