Using a grinder attachment for a stand mixer or pulsing in a food processor, coarsely grind the tenderloin meat. (Your supermarket's butcher department can also do this for you.) Transfer to a bowl. Add the parsley, onion, egg yolk and salt and pepper to taste. Form 4 patties.
Heat a griddle or large, heavy skillet over medium-high heat. Add the butter; when it foams, add the burgers and cook 6 to 8 minutes for rare to medium-rare, turning once. For cheeseburgers, top with the cheese for the last minute of cooking.
Serve the burgers on the rolls. Top with the horseradish sauce and ketchup.