Slice the frankfurter into rounds and roughly chop the kimchi.
Bring a small saucepan of water to the boil. Cook the instant noodles in the boiling water for 2 minutes, or until al dente (just firm to the bite). Drain in a colander, then tip into a bowl of cold water to refresh. Drain again and set aside in the colander.
Place the large non-stick frying pan over a medium heat, then add the butter and allow it to melt and heat up. Add the frankfurter slices and kimchi, and saut&; (fry gently) for 2 minutes.
Add the noodles and the noodle flavour sachets. Toss everything to combine, then use a pair of kitchen scissors to chop the noodles up a little.
Place the small non-stick frying pan over a medium heat, add the vegetable oil and let it heat up. Don't skimp on the oil - it sounds like a lot, but it really helps to get the egg super-crispy. Tap the egg on the bench to crack the shell and break it into the hot oil. Fry the egg until the edges start to crisp up. To help the egg white cook, tip the pan and spoon hot oil over the white. When the egg white is almost set, carefully sprinkle the grated cheese around the outside of the egg and cook until the cheese is crispy and golden but the egg yolk is still soft. Place the noodles on a plate and top with the fried cheesy egg to serve.