Slice each leek in half lengthwise, and each half into ½-inch-thick half-moons. Place in a large bowl of cold water and swish. Drain and pat dry.
Melt the butter in a large saucepan over medium heat. Add the leeks and cook for 3 minutes.
Add the potatoes, wine, and broth and bring to a boil.
Reduce heat, cover partially, and simmer until the potatoes are tender, about 15 minutes.
Remove from heat. Add 1 teaspoon salt, ¼ teaspoon pepper, and the watercress. Stir until the watercress wilts.
Meanwhile, slice the biscuits in half. Place the ham on the bottoms and sandwich with the tops. Ladle the soup into bowls. Serve with the ham biscuits.
