Put a large pan of salted water on to boil. Peel and grate the garlic and ginger. Trim the roots and the top 3cm off the spring onions. Finely slice the spring onions, white and green parts, on the diagonal. Separate the pak choi leaves. Finely shred the thicker stalks and then slice any larger leaves.
The water should be boiling by now. Add the udon noodles to the pan. Simmer for 8 mins till the noodles are tender.
While the noodles simmer, put a wok or deep frying pan on a high heat for 2 mins. Add 1 tbsp oil and the beef. Stir fry for 3-4 mins till the beef is browned all over. Lift the beef out of the pan and pop it on a plate.
Add ½ tbsp oil, the garlic, ginger and most of the spring onions (keep a pinch of the green parts back for garnishing). Stir fry for 30 secs till the pan smells fragrant. Add the pak choi stems to the pan and stir fry for 2-3 mins.
Add the beef back to the pan. Pour in 2 tbsp balsamic vinegar and add the sesame seeds. Drop in the pak choi leaves. Toss to mix and just wilt the pak choi leaves, around 2 mins.
The noodles should be cooked by now, so drain them and add them to the wok. Pour in 1 tbsp sesame oil and toss everything together to mix.
Divide the stir-fry between 2 warm bowls and garnish with the last of the spring onions to serve.