Peel the carrots and grate in a food processor or by hand.
Combine the carrots, rice vinegar, sugar, and clam juice in a large saucepan. Bring the liquid to a boil over medium-high heat. Break the noodles in half and add them to the pan. Season to taste with salt and pepper. Cover and simmer until the noodles are almost cooked through, about 5 minutes.
Meanwhile cut the fish into 2-inch pieces.
Add the fish to the pan, cover, and simmer until the fish and noodles are cooked through, about 5 minutes more.
Meanwhile trim and thinly slice the scallions.
When the fish is cooked, stir in the scallions and sprinkle the top with sesame seeds if you wish. Serve hot.
If you can’t find buckwheat noodles, substitute vermicelli or capellini. The dish won’t taste as distinctive, but it will still be good.