Boil the potatoes until fork tender. Strain. Mash the potatoes with the butter and milk. Season to taste and set aside to cool
Stir the egg and breadcrumbs into the potatoes. Form patties. Fry on each side in a little olive oil to brown. Remove from heat.
Cook the beef on high heat to brown. Remove from pan.
Add the onions, carrots, and celery and cook about 10 minutes to soften.
Add the garlic and ginger and cook 2 minutes.
Add in the stocks, miso, vinegar, soy, and chile paste. Return the beef to the pot. Stir and simmer for 30 minutes.
Add the corn and spinach and stir to combine.
In each bowl, add some of the cooked ramen. Ladle the soup over the noodles. Top with an egg and the potato cake and serve.