Skin the sole. Cut off the small fins on the sides of the sole and the head. Rinse off any impurities under cold, running water.
Pat dry with a clean towel. Season with salt and pepper.
Brown the butter in a non-stick pan over high heat. Place the fish in the pan and constantly add knobs of butter, making sure it doesn't burn. Brown both sides of the fish. Remove the fish from the pan and keep warm.
Sauce
Return the pan containing the sole to the heat. Add a knob of butter until it bubbles. Remove the pan from the heat. Finely chop the parsley. Season with salt, pepper, the juice of half a lemon, and the parsley. Place the North Sea sole in the pan.
Potatoes
Preheat the oven to 140°C (275°F). Fill a baking dish with the salt. Wash the potatoes with their skins and place them on the salt. Bake for 40 minutes. Peel the potatoes and mash them with a little butter and olive oil. It's quicker to boil the potatoes in their skins for 20 minutes, or until they are thoroughly cooked. Then peel and mash as described above.