If using frozen octopus, thaw it fully in the refrigerator, then rinse and pat dry. If using fresh whole octopus, rub it firmly with coarse salt for 2 minutes to remove surface sliminess, then rinse well.
Cut the octopus into bite-size pieces, about 4 cm / 1 1/2 in, tentacles and body both.
Whisk the gochujang, gochugaru, soy sauce, sugar, garlic, ginger, and sesame oil together in a small bowl until smooth.
Heat the neutral oil in a wide pan or wok over high heat until it shimmers.
Add the onion and carrot and stir-fry for 2 minutes, until the onion turns translucent at the edges.
Add the octopus and the sauce together, all at once, and toss constantly over high heat.
Cook for 3 to 4 minutes only, tossing the whole time, until the octopus turns opaque and firms up slightly but is still tender when you bite into a piece. Pull it off the heat the moment it looks done — a few seconds past that point and it turns rubbery.
Add the cabbage and scallions in the last minute of cooking, tossing just until they wilt slightly but keep some crunch.
Scatter sesame seeds over the top and serve immediately with a bowl of rice.