Whisk together vinegar, shallot, honey, and salt in a small bowl. Add oil in a slow, steady stream, whisking until incorporated. Stir in 1 tablespoon chopped mint. Set aside.
Remove and discard seeds from 1 half of each melon; cut each into 2-inch-wide radial spokes, about 6 slices each. Reserve remaining melon halves for another use.
Using a sharp knife, follow the natural curve of the melon to remove the rind. Cut slices in half.
Arrange melon pieces and ham slices on a platter. Drizzle vinaigrette over top; sprinkle with black pepper and remaining 1 tablespoon mint.