Preheat the oven to 200°C, fan 180°C, gas 6. Place the potatoes in a medium saucepan and cover with water. Bring to the boil and cook for 7-8 mins until tender, then drain well.
Return the cooked potatoes to the saucepan, add the yogurt and mash together well until as smooth as possible. Flake in the fish and, using a fork, stir this into the potato with the chives and lemon zest. Cover with cling film and place in the fridge for 20 mins until firm and easy to handle.
On a floured surface, shape the mixture into a large round cake about 2cm thick and use a fish-shaped cutter to cut out fish cakes, reshaping as needed. You should end up with about 14 fish.
Put the egg in a shallow bowl and the breadcrumbs in another shallow bowl. Dip the fish shapes in the egg, then in the breadcrumbs and arrange on a baking paper-lined baking tray. Bake in the oven for 12 mins or until golden.
Serve the fishcakes with the carrot and cucumber batons.