For the tots, boil potatoes until tender. Drain and pass through a ricer into mixing bowl. Combine with grated onion, flour, egg, minced rosemary and season with salt and pepper. On a floured surface, roll into 1-inch thick logs and cut into tots.
Heat frying oil in a countertop fryer to 350F or heat a few inches of oil in a large Dutch oven. Fry tots in small batches when burgers are ready to serve. Season cooked tots with a little extra salt.
For the burgers, with meat grinder attachment on a stand mixer, coarse-grind meat if you didn't get it already ground. Transfer ground meat to a mixing bowl and add parsley, onions, salt, pepper and egg yolk. Form 4 patties. Heat a cast-iron pan or large heavy skillet over medium-high heat. Add butter and when it foams, add burger patties. Cook 6-8 minutes for rare to medium-rare, turning once. Add cheese in last minute for cheeseburgers, tenting with foil to melt. Serve on brioche rolls with tots alongside.