Slice the green onion diagonally into thin pieces.

Heat water, soy sauce, and grated ginger in a pot over medium heat until boiling.

Add udon noodles and sliced green onion to the pot and cook according to the noodle package instructions.

Transfer the noodles and green onion to a serving bowl. Mix cornstarch and water in the pot with the remaining broth.

Heat the broth until thickened, then turn off the heat and slowly pour in the beaten egg while stirring gently.