Heat oil in a large pot over medium heat. Add in the grated carrots and onion and cook for several minutes, until the onion is translucent.
Make a well in the centre of the mixture and add in the minced meat. Cook until browned, gradually stirring it into the carrot and onion mixture. This will take several minutes. Make a well again and add in the mushrooms until they are heated through. Throw in the dried oregano and dried basil. You can use fresh ones if you have them on hand.
Make a well in the centre of the mixture again, dump in the tomato paste and let them heat up. Throw in the garlic and mix it all up.
Make a well again, this time adding in the red wine/chicken stock and the soy sauce, stirring to combine with all the other ingredients.
Add in the canned tomatoes and bring the mixture to boil then stir in the milk. Simmer for 30 minutes to 2 hours, however long you need. Season with salt and black pepper. When done, start the assembly of the lasagna.
Preheat the oven to 180 degrees Celsius. Lightly grease an ovenproof dish.
Pour half cup of the meat sauce on the bottom of the ovenproof dish. Spread it out. Lay the lasagna sheets on top then top with more meat sauce, making sure that the lasagna sheets are completely covered in meat sauce. Sprinkle some of the mozzarella cheese and cheddar cheese on top. Repeat the layers (lasagna sheets, meat sauce, cheeses, lasagna sheets, meat sauce...), ending with the cheese layer. Sprinkle Parmesan cheese evenly over the top and send the whole dish into the oven for 30 to 40 minutes, until the pasta is cooked and the top is browned.
Remove the dish from the oven and set it aside for 5 minutes before digging in.