Combine goat cheese and 1 tablespoon olive oil in a medium bowl. Season with salt and pepper.
Heat grill to medium-low. Turn dough out onto a floured surface and divide into 4 pieces. Roll each piece into a 7-inch-long oval and brush both sides with olive oil. Place flatbreads directly on grill grates and cover grill. Cook, checking every 2 minutes, until dough bubbles and stiffens and flatbreads have visible grill marks on the bottom, about 4 minutes. Flip flatbreads, cover grill, and cook 1 minute. Remove from grill.
Divide goat cheese mixture among the flatbreads and spread to coat the surface. Place 2 to 3 slices of prosciutto on each flatbread. Return flatbreads to grill, cover, and cook, checking every 2 minutes, until cheese is softened, ham is crisp at the edges, and dough is fully cooked, 3 to 4 minutes. Remove from grill.
Lay 2 to 3 slices of melon on each flatbread. Garnish with mint leaves, a drizzle of olive oil, a dusting of flaky salt, and black pepper.