Heat Lemon Garlic in a large nonstick skillet over medium heat. Add potato noodles and sprinkle with salt and pepper. Cover and cook, stirring occasionally, until tender and cooked through, about 12 minutes.
Transfer noodles to a bowl and stir in beaten eggs. Toss until well blended. Using kitchen shears, snip noodles in bowl to about 1-2-inches in length.
Place same skillet back over medium-high heat and add another 1 tablespoon Lemon Garlic to pan. Add 1/3 cup potato noodles to pan and flatten. Cook potato noodle “buns” until browned, about 5-6 minutes, then flip, flatten and cook until browned. Remove potato buns to a platter and set aside.
Meanwhile, divide beef into 4 pieces and shape into patties with an indent in the center; sprinkle both sides with salt and pepper.
Heat large cast iron skillet over medium-high heat and cook burgers until well-browned, about 3-4 minutes, then flip and cook another 1-2 minutes for medium. Remove burgers to a plate and let rest 5 minutes.
While burgers are resting, heat remaining 1 tablespoon Lemon Garlic in a nonstick skillet over medium heat. Add eggs, cover and cook until whites are set and yolk is still runny, about 3-4 minutes.
Stir together Mayonnaise and Ketchup. Spread over each potato “bun”. Place 1 burger on a potato “bun” and top with lettuce, a few onion rings, egg, and another bun.