Place cucumber and julienned carrots in small mason jars, cover with vinegar and let sit at least 30 minutes.
Mix together ground turkey and furikake until uniform. Make 4 large, but very thin, patties.
Set a large frying pan at medium heat, and place burgers in it, cooking for 4-5 minutes on each side, until cooked through.
While the burgers are cooking, in another non stick pan on low heat melt butter, then crack all four eggs into the pan. Slow cook the eggs sunny side up until all the whites are cooked and the yellows are still runny, remove from heat.
Assemble the burgers, with gochujang and mayo on the bottom bun, then the burger, then place the rest of the ingredients in this order if you choose: pickled cucumber, kimchi, nori, egg, pickled carrots and bean sprouts.