Beat egg and water in a bowl; add ground beef, bread crumbs, salt, and black pepper to eggs and mix thoroughly. Form ground beef mixture into bite-size meatballs; place meatballs into a large baking dish.
Bake in the preheated oven until meatballs are browned and no longer pink inside, 25 to 30 minutes.
Mix ketchup, beef bouillon, and vinegar in a large saucepan or Dutch oven.
Combine brown sugar and cornstarch in a small bowl and stir brown sugar mixture into ketchup mixture. Bring to a boil, reduce heat to low, and stir until sauce is thickened and bubbling and brown sugar has dissolved, about 5 minutes.
Add meatballs to sauce, stir until coated, and serve.