Special equipment: 13 by 9-inch baking pan
Preheat the oven to 350 degrees F.
In a medium pot, heat the olive oil and add the onions and garlic. Season with salt and pepper, to taste. When it turns light brown, add the thyme, bay leaf and tomatoes. Simmer for 20 minutes, then taste for seasoning. Add more salt and pepper, if desired. Remove and discard the thyme and bay leaves. Stir in the basil leaves and shut off the heat to allow the sauce to "rest."
Pasta: Bring the water to a boil over medium heat. Add the salt and the lasagna sheets. Cook for 4 minutes, drain and rinse under cold water. The pasta should still be very firm to the touch. Separate the sheets carefully so they don't stick together.
Divide the tomato sauce in half and reserve half for the Bolognese and half for the lasagna.
Assemble the lasagna:
Spoon a thin layer of the sauce in the bottom of the baking pan. Arrange a layer of pasta sheets over the sauce. Sprinkle some of the mozzarella and Parmesan over the pasta and another thin layer of sauce. Repeat the layering process 2 more times. It is important there be remaining cheese and sauce for the top. Cook's Note: I love to get a corner piece from the pan and pick at the crispy edges of the top layer. When you finish, there should be 4 layers of pasta and 5 layers of filling.
Cover the dish tightly with aluminum foil and put it in the center of the oven. Bake for 45 minutes. Raise the temperature of the oven to 450 degrees F and remove the aluminum foil. Bake until the top browns slightly, about 10 to 15 minutes. Remove the pan from the oven and let it cool for 15 minutes or so before serving.
Make the Bolognese while the lasagna is baking:
In a large wide pot over medium-high heat, add the olive oil. Add the ground beef and cook until brown, about 4 to 5 minutes, stirring with a wooden spoon to break up the pieces. Season with salt and pepper, to taste. Remove the beef from the pot and set aside. Set the pot back over the heat and add a little more olive oil. Add carrots, onion and celery. Cook until browned, about 5 minutes. Add the beef back to the pot, then add the reserved 1/2 recipe of tomato sauce. Cover slightly and simmer for 20 minutes.
When done, finish with milk, taste, season and serve, spooned over the lasagna.