Add the olive oil to a large pot along with the onions, garlic and carrot. Cover with a lid and cook over medium low heat until the onions are soft and translucent. The moisture released by the vegetables should keep them from burning, but if they start burning, turn down the heat and add a little water.

Once the onions are cooked, remove the lid and turn up the heat, sautéing until the mixture is ⅓ the original volume and starting to caramelize.

Push the vegetables to the sides of the pan, turn up the heat to medium-high and add the ground beef and pork. Use a spatula to break up the clumps.

Add the milk and boil, continuing to break up the clumps until the beef is cooked and there is no liquid left.

Add the wine and boil until most of the liquid has evaporated and there is no smell of alcohol remaining.

Add the tomatoes, using your hands to crush them into small pieces, and then add the tomato paste, salt, oregano, and black pepper.

Use a microplane to shave the dried shiitake into the pot (you can also put the dried mushroom in a spice grinder and powderize).

Simmer the sauce over medium low heat until it is thick (30-40 minutes).

Generously salt a large pot of water and add a teaspoon of olive oil. Bring the water to a boil, then cook the pasta 3 minutes less than what the package directions say. My pasta said 9 minutes so I cooked it for 6.

Place the oven rack in the top position and preheat to 350 degrees F (180 C).

Put the Mozzarella and Gruyere in a bowl and toss to distribute evenly.

To construct the lasagane, put down a layer of sauce that's just thick enough to cover the bottom of the dish.

Sprinkle with cheese.

Cover with the parboiled pasta, cutting the pasta as necessary to fill in the spaces. Repeat so that you have 3 layers of pasta.

Once you have 3 layers of pasta, finish by spreading the remaining meat sauce on top of the last layer of pasta. Cover with enough cheese so that you cannot see any sauce underneath.

Put the lasagne on a baking sheet to catch any spills, then place it in the oven. Bake until the sauce is hot and bubbly and the cheese has browned on top. If the sauce has been bubbling for a while and the cheese is not getting brown, turn up the heat to 450 degrees F (230 C) to finish it off.

When the lasagne is done, take it out of the oven, garnish with minced parsley, and let it rest for 10 minutes before cutting and serving.