Lasagne al Forno - reloaded!

About 2 years ago, I prepared my first lasagne with homemade pasta, later again as a Daring Bakers' challenge. In fact the last time I cooked lasagne was that mentioned challenge. Meanwhile, I discovered that I indeed like cooked tomatoes, so I reinvented my Ragù alla bolognese. Nowdays, I cook three types of ragù and this is one of them and it ended up in a lasagne for lunch today.

Ingredients:
2 tablespoon olive oil
1 onion
1 carrot
1 celery stalk
2 cloves garlic
2 pinches of dried oregano
500 g beef, minced
1 tablespoon tomato puree
1 bay leaf
1 clove
5 stalks of basil
50 ml red wine
300 g tinned tomatoes (preferably San Marzano)
60 ml milk
salt, pepper
For the Béchamel:
50 g butter
50 g flour
600 ml milk
nutmeg
salt, pepper
For the pasta:
200 g flour
2 eggs
1 tablespoon olive oil
salt
mozzarella
parmesan

Preheat oven to 200°C. Heat olive oil, add grated onion, carrot and celery. Peel garlic and throw it to the pot and fry it all together with the bay leaf (stick clove into the leaf), oregano, salt and pepper. As soon as the onion is soft, add beef and stir. Add tomato puree and cook for a minute and fry until the meat is brown. Add wine, reduce and add tomatoes. Simmer for about 15-20 minutes. Stir in the milk, bring to the boil and set aside. For the Béchamel melt butter, stir in flour and add the milk while whisking to avoid lumps. Cook for about 2-3 minutes, or until it thickens. Season with salt, pepper and fresh nutmeg. Remove from the heat. Knead a smooth pasta dough using flour, eggs, salt and olive oil and let it rest for 30 minutes. Cut it in 8 pieces and roll it out as thin as possible with your pasta machine. Remove garlic cloves, bay leaf and basil from the ragu and put a spoonful of ragu in an ovenproof dish, then add some béchamel, put 2 long sheets of lasagne on it, again ragu, béchamel, parmesan and so on. On the last layer put some béchamel, mozzarella, parmesan and some chunks of butter. Bake for 25-30 minutes. I suggest to let it stand for 10 minutes before serving.