In a large saucepan, melt 4 tablespoons butter over medium-low heat. Add the scallions and cook for 1 minute. Add the potatoes, chicken broth, wine and 1/2 teaspoon salt; bring to a boil. Lower the heat, cover partially and simmer until the potatoes are tender, 20 to 25 minutes.
Using a blender, puree the soup. Return to the pot and stir in the heavy cream; season with salt and pepper. Cover and keep warm.
Rinse each sheet of matzo under running water, then break into small pieces and add to a large bowl. Stir in the eggs and milk and season with salt and pepper.
In a large nonstick skillet, melt 1 tablespoon butter over medium heat. Transfer the matzo mixture to the skillet, patting it down in an even layer. Cover and cook until golden-brown on the bottom, about 5 minutes.
Hold a flat cookie sheet over the skillet and carefully invert the matzo pancake onto it. In the skillet, melt the remaining 1 tablespoon butter over medium heat. Slide the pancake into the skillet and cook until golden-brown on the other side, about 5 minutes. Slide onto a cutting board and cut into 8 wedges.
Sprinkle the parsley on the soup and serve with the fried matzo dippers.