Preheat the oven to 200 °. Cook the potatoes in salted water until they are done, drain and mash them very roughly, i.e. you should still be able to see isolated pieces. Then use a wooden ladle to stir in milk, 2 lightly heaped tablespoons of butter and 50 g of Parmesan. Then try, pepper and possibly add a little more salt. Finally fold in the chives, pour everything into the unbuttered casserole dish, sprinkle 2 lightly heaped tablespoons of butter in small flakes over it, sprinkle the remaining 20 g Parmesan on top and let it gratin in the oven for 20 minutes. Any leftovers can be fried in a pan the next day and served with a fried egg.
Now season the fish all around and drizzle a little lemon juice over it. Heat the butter in a larger, high pan, stir in the flour until it foams slightly, then deglaze with the milk and then with the wine while stirring vigorously. Keep stirring until you have a creamy sauce. Then let everything simmer gently over a medium heat for 2 minutes and finally season with pepper, salt and a pinch of sugar. Place the pieces of fish in the sauce so that they are completely covered. Then keep the medium heat on for another 5 minutes, but then turn it off and let the fish simmer for another 3 minutes.
While cooking the fish, cook the carrot grated in salted water. This takes a maximum of 2 - 3 minutes. When they are soft but still have a slight bite, drain and refine with a tablespoon of butter. If it was spirals, then it is intended that they break up into smaller pieces. For coarsely grated carrots, keep an eye on the cooking time. You could be soft faster.
Then all 3 components should be ready at the same time, and it can be served. Good Appetite...........