Bring a pot of water to a boil and cook the sweet potato noodles for 6 minutes until translucent and chewy.
While noodles cook, whisk together the miso paste, gochujang, soy sauce, maple syrup, and lime juice in a small bowl until smooth.
Heat the sesame oil in a non-stick skillet over medium-high heat.
Place the salmon patties in the skillet and sear for 4 minutes on each side until a golden-brown crust forms and they are heated through.
Remove the patties from the skillet and set aside; add the thinly sliced red onion to the same skillet and sauté for 2 minutes.
Drain the noodles and add them to the skillet with the onions, then pour the miso-gochujang sauce over the top.
Toss the noodles for 1 minute until they are well-coated and glossy, then transfer to a serving bowl.
Top the noodles with the salmon patties, kimchi, and pickled carrots, finishing with a drizzle of chili oil for a spicy kick.