Peel the carrots, parsnips and sweet potatoes. Grate thin strings of the vegetables with a peeler, cheese slicer or spiral cutter.
Heat half the butter in a frying pan. Season the fish with salt and pepper and fry the fillets briefly over high heat.
Heat the rest of the butter in another pan and fry the vegetable tagliatelle for a few minutes until al dente. Divide the tagliatelle between two plates and place the fish and pesto on top.