Slice each leek in half lengthwise, then slice each half into 1/2-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 and add the salt, pepper, and watercress. Stir until the watercress wilts.
Meanwhile, slice each biscuit in half. Place the ham on the bottoms and sandwich with the tops. Ladle the soup into individual bowls and serve with the ham biscuits.
Tip: To make a creamier soup, puree half of it in a blender or food processor just before serving. Return the puree to the pot to rewarm it.