Let 5 1/2 ounces fried fish cake sheets (3 to 4) and 10 ounces frozen rice cakes sit at room temperature until thawed. (Alternatively, place the rice cakes in a medium bowl, cover with cold water, and set aside to thaw for 10 minutes. Drain before using.)
Cut the greens from 3 medium scallions crosswise on a slight diagonal into 1/2-inch-wide pieces (save the whites for another use). Cut the fish cake sheets crosswise into 1-inch-wide strips.
Place 3 1/2 cups water, 5 dried anchovies, and 1 (about 4×1-inch) piece kombu in a large saucepan and bring to a boil over high heat. Boil for 3 minutes. Remove the anchovies and kombu with a slotted spoon and discard.
Add 1 block instant ramen noodles (discard the seasoning packet) to the pot and cook, stirring to separate the noodles, for 1 minute. Using a slotted spoon or tongs, transfer the noodles to a bowl.
Add 2 tablespoons gochujang, 2 tablespoons oligo or light corn syrup, and 1 tablespoon gochugaru chili powder to the pot, and stir until fully dissolved. Add the rice cakes and fish cakes and cook on medium-high heat, stirring often, until heated through and the flavors start to meld, about 2 minutes.
Return the noodles to the pot and cook, stirring occasionally, until al dente, about 1 minute. Turn off the heat. Taste and season with kosher salt as needed. Top with the scallions and 1 tablespoon roasted sesame seeds. Divide the tteokbokki between bowls or serve straight from the pan; the sauce will thicken slightly as it sits.