Using a sharp knife, cut off rinds from watermelons. (You should have a total of 2 pounds peeled fruit.) Quarter each melon, and then cut into 3-inch-long, 1/4-inch-thick slices. Arrange slices on a serving platter.
Using a vegetable peeler, shave paper-thin slices of ricotta salata over watermelon. Sprinkle with salt and mint, and serve immediately.
This salad looks beautiful when made with both red and yellow watermelon, but is just as lovely when only one color is used. Ricotta salata, a salted, dry Italian sheep's milk cheese available at cheese counters and specialty-food stores, is ideal here, but feta may be substituted.