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 350°F 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.