Peel and slice the carrots, cook in salted water for about 15 minutes and drain. Wash and clean broccoli, cut into florets, cut stems into slices, cook in salted water for about 10 minutes and drain.
Puree the carrots and broccoli separately with the cutting stick of the hand mixer. Allow to cool slightly and mix with one egg, one egg yolk and 1/2 cup of crème fraîche. Season to taste with salt, pepper and nutmeg.
Heat the fat in a pot and sauté the flour in it. Add milk while stirring and boil thoroughly. Season to taste with lemon juice, salt and pepper. Put half of the sauce into an ovenproof dish.
Cover with a layer of lasagne sheets. Spread broccoli mixture on top, then layer pasta, carrot mixture, noodles and the rest of the béchamel sauce. Crumble the cheese on top and bake in the preheated oven (electric: 200 °C/ gas: level 3) for about 45 minutes.