Preheat the oven to 375 degrees F.
In a large saucepan over medium-high heat, add the oil.
Add the beef, the sausage, the onions, the carrots, the celery, and the salt to the heated oil and cook, while crumbling the beef and the sausage, until the veggies are soft and the meat is no longer pink, about 10 minutes.
Add the garlic, the oregano, and the red pepper flakes to the meat mixture and cook until they are fragrant, about 1 minute.
Add the tomato paste to the meat mixture and cook, about 2 minutes.
In a blender, add the tomatoes and blend until they are puréed.
Pour the tomato purée into the meat mixture and bring to a boil.
Reduce the heat to low and simmer the meat sauce, stirring occasionally, until it is reduced and has slightly thickened but is still very loose, about 15 minutes.
Transfer the saucepan from the heat.
In a large bowl, stir the ricotta, the milk, the eggs, 1 cup of the Parmigiano-Reggiano, 2 1/2 cups of the mozzarella, 3/4 cup of the parsley, the salt, and the pepper together.
Evenly spread 2 cups of the meat sauce over the bottom of a 9x13-inch baking dish, preferably ceramic or enameled cast-iron.
Arrange 3 of the lasagna noodles over the sauce, spacing them out slightly so there is room for them to expand while baking.
Spread 2 cups of the ricotta mixture over the noodles.
Arrange 3 of the noodles over the ricotta mixture.
Repeat the layers using 2 cups of the meat sauce, 3 of the noodles, 1/2 of the remaining ricotta mixture, and then 3 more of the noodles, in that order.
Add a final layer using 2 cups of the meat sauce, 3 of the noodles, the remaining ricotta mixture, and then the remaining 3 noodles, in that order.
Evenly spread 2 cups of the meat sauce over the noodle layer and sprinkle with the remaining Parmigiano-Reggiano and the remaining mozzarella.
Loosely cover the baking dish with parchment-lined foil and place it on a rimmed baking sheet.
Bake the lasagna until the noodles are tender but still al dente, about 40-45 minutes.
Discard the foil and bake the lasagna until it is bubbly, slightly puffed along the edges, and the top is golden-brown in some places, about 20-25 minutes.
Transfer the baking dish to a wire rack and let the lasagna cool slightly, about 20 minutes.
Serve the lasagna garnished with the extra parsley.