For the white remoulade: In a food processor, fitted with a metal blade, combine the egg, mustard, lemon juice, garlic and horseradish. Process until smooth. With the machine running and in a steady stream, add the oil until the mixture is thick. Season with salt and pepper. Cover and refrigerate until chilled.
Spread each side of the bread with the butter. Place the bread on the griddle and cook for 2 to 3 minutes on each side or until golden. Remove from the griddle. Spread each piece of the bread with the white remoulade. Mound about 1/3 of a pound on the meat on each piece of bread. Lay 2 slices of the cheese on top of each mound of meat. Place each piece of bread on a baking sheet. Place in the oven and bake until the cheese is bubbly, about 6 to 8 minutes. Fry the potatoes in a little hot oil in a saute pan until golden brown, about 3 to 5 minutes, stirring occasionally. Remove and drain on paper towels. Season with salt and pepper. To serve, place 2 open-faced sandwiches on each plate. Garnish each with a pile of potatoes and parsley.