2 medium zucchini
1 teaspoon salt
2 (3-inch-wide) wedges melon, seeded and rind removed
1 whole smoked chicken breast (1 lb), skinned and thinly sliced crosswise with a knife
1 (1/2-lb) piece Parmigiano-Reggiano
1/4 cup packed fresh mint leaves, cut crosswise into thin shreds
1/4 cup extra-virgin olive oil
1/4 cup fresh lime juice
Cut zucchini crosswise diagonally into 1/8-inch-thick slices 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.
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.