Preheat the oven to 450 degrees F. Put the corned beef in a food processor and pulse until finely ground; remove to a medium bowl. Add the bread to the food processor and pulse until fine; add to the corned beef. Add the ground beef, egg, milk, onion, 3/4 cup cheese, 1/4 teaspoon salt and a few grinds of pepper. Mix well with your hands and transfer to a rimmed baking sheet. Form into four 3/4-inch-thick oval patties.
Bake the patties until lightly browned, firm and cooked through (a thermometer inserted into the centers should register 160 degrees F), 20 to 23 minutes. Evenly top the meatloaves with the sauerkraut and remaining 1/2 cup cheese. Return to the oven and bake until the cheese is melted, about 3 minutes.
Transfer the meatloaves to plates and top with Russian dressing and parsley. Serve with potato chips and pickles.