Cut zucchini crosswise diagonally into 1/8-inch-thick slices using slicer and transfer to a colander set over a bowl. Sprinkle with salt, tossing to coat, and let stand 5 minutes, then rinse under cold water. Arrange in 1 layer on paper towels and pat dry. Cut melon wedges lengthwise into 1/8-inch-thick slices using slicer. Divide melon, zucchini, and chicken among 4 plates. Shave about one fourth of cheese into curls with a vegetable peeler and divide curls and mint among the 4 plates. Drizzle each plate with oil and lime juice. Season with salt and pepper. Serve immediately.