Rise rice thoroughly in a mesh colander. Bring water and rice to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender and water has been absorbed, 20 to 25 minutes. Set rice aside.
Heat 1 tablespoon oil over medium-high heat in a large wok. Add ground beef and a few pinches of salt. Cook thoroughly, until meat is crumbly and no longer pink, 5 to 10 minutes. Transfer cooked beef to a bowl, leaving juices in the wok.
Add potato, carrot, and onion to the wok. Cook and stir over medium-high heat until potato seems halfway cooked, about 7 minutes. Add zucchini and a few dashes of salt and continue cooking until potato is tender, about 5 minutes more. Return cooked ground beef to the wok and mix with vegetables. Sprinkle black pepper into the mixture and stir to combine.
Add remaining 1 tablespoon oil and cooked rice to the wok with the beef mixture. Mix together, trying not to make the rice cakey, and cook for about 5 minutes. Add sesame seeds and mix again; cook about 5 minutes more. Season with additional salt, if desired.
Whisk eggs in a bowl. Heat a medium-sized frying pan over medium-low heat. Cook eggs in 4 batches to make thin fried eggs the size of the frying pan, flipping each one time, about 2 minutes per egg.
Serve by placing fried rice onto a plate in a circular shape and place 1 fried egg directly on top, covering rice completely. Apply ketchup over egg in a zigzag or other desired pattern and serve. Eat ketchup, egg, and rice together in 1 spoonful.