Put water in a pot and bring it to a boil. Once boiling, reduce the heat to low, add bonito flakes, and let it simmer for 3 minutes.
Turn off the heat and strain the mixture through a sieve lined with paper towels or a cloth (such as cheesecloth). Alternatively, if you don't mind some fine bonito flakes remaining, you can simply use a fine-mesh strainer. Bonito dashi (udon broth) is now ready.
Cut the Japanese leek into ⅕-inch (5 mm) thick diagonal slices. Thinly slice the green onion diagonally or into rings. Cut the beef into bite-sized pieces.
Return the bonito dashi to the pot, add the leeks, beef, and the seasonings (mirin, sugar, and soy sauce), and bring it to a boil. Once boiling, reduce the heat to low and simmer for about 3 minutes, skimming off any scum.
Turn off the heat, add curry roux to the pot, and stir until dissolved. In a small bowl, mix starch with water, then gradually add the mixture to the pot. Reheat the pot and simmer briefly to eliminate any powdery texture from the starch.
If using fresh, dried, or pre-cooked udon noodles: Boil the udon noodles according to the package instructions. Once cooked, drain them in a colander.If using frozen udon noodles: Lightly sprinkle water over the udon noodles. Then, place them on a microwave-safe plate, cover with plastic wrap, and microwave on medium power (500W) for about 4 minutes. (Since this method can cause uneven heating, it is best to microwave one serving at a time. You can also boil them like the other types of udon noodles.) Place the noodles in bowls and pour the curry soup over them. Loosen the noodles with chopsticks, then top with the green onions.