In a pot melt the butter, add the flour and then pour the milk stirring constantly to avoid lumps. Season with salt and a little nutmeg then bring to boil. cook for only a couple of minutes. Set aside.
Heat a pan and sauté chopped carrot, onion and celery in a little olive oil. Add the mixed minced beef and/or pork and a bay leaf; stir. Once the meat has reached a brown colour, pour in the wine and let it reduce. Add the tomato puree, cover with water and let cook for at least 30 minutes. Season with salt and pepper, remove the bay leaf and set aside.
Clean the mushrooms, slice them finely and cook in a non-stick pan over medium heat. Set aside.
Grease a baking dish with a little bit of butter. Spoon a layer of béchamel, then Bolognese sauce over the bottom of the dish. Top the first layer with lasagne sheets, then cover it with some sliced mushrooms and sprinkle with grated Parmesan.
Repeat this procedure 5 times, altering layers of Bolognese sauce, lasagne, mushrooms and béchamel. Make sure to leave enough béchamel sauce to cover the top layer completely.
Bake in a preheated oven at 220 C for 25-30 minutes. Let the lasagne rest for ten minutes before serving.