Preheat the oven to 400 degrees, or preheat the grill to medium.
Combine the strawberries, vinegar, and a few grinds of pepper in a medium, no-reactive mixing bowl and let stand at room temperature for at least 5 (and up to 30) minutes.
Brush the flatbread lightly on both sides with olive oil and sprinkle with salt to taste. Place on a baking sheet and heat in the oven 2 minutes to soften; or place flat-breads directly on the heated grill for 2 minutes to soften and toast slightly. Remove, cut in quarters and return to baking sheet.
Slice the mozzarella into 1/2-inch-thick slices. Layer 3 to 4 mozzarella slices on the tasted pita quarters, and return to the oven until the cheese is puffy, about 2 minutes; or, place pita quarters directly on the grill and close the lid until the cheese is puffy, about 2 minutes. The cheese may get a few toasty brown spots, but watch carefully to prevent over-browning.
Finely chop the basil leaves and stir gently into the strawberries. When the pita quarters come out of the oven or off the grill, spoon the strawberry mixture over the cheese and serve immediately.