Stir the cornstarch, water and sesame oil in a small bowl until the mixture is smooth.
Heat the broth and soy sauce in a 3-quart saucepan over medium heat to a boil. Stir the cornstarch mixture in the saucepan. Cook and stir until the mixture boils and thickens.
Reduce the heat to low. Gradually stir the eggs into the saucepan. Remove the saucepan from the heat. Divide the spaghetti, ham and green onions among 8 serving bowls. Spoon about 1 cup broth mixture into each. Serve immediately.