This rich baked pasta is made with layers of tender short ribs, creamy basil-flecked cheese and sweet tomato sauce.
Make the Ribs: In a large Dutch oven or heavy-bottomed stock pot, heat the oil over medium-high heat. Season the ribs with the salt and pepper. Add the ribs to the pan and cook for about 4 minutes each side until brown. Remove the ribs and set aside. Add the onion, garlic and rosemary. Season with 2 teaspoons salt and 1 teaspoon pepper. Cook for 5 minutes until the onions are translucent and soft. Increase the heat to high. Add the wine and scrape up the brown bits that cling to the bottom of the pan with a wooden spoon. Add the beef broth and ribs to the pan. Bring the mixture to a boil. Reduce the heat to a simmer, cover the pan and cook for 2½ to 3 hours until the meat is very tender. Remove the ribs and set aside until cool enough to handle, about 20 minutes. Discard the bones and cooking liquid. Using two forks, shred the meat into 2-inch-long pieces (to yield approximately 2¼ cups shredded meat).
Make the Filling Mixture: In a medium heavy-bottomed saucepan, bring the milk and cream to a simmer over medium heat. Reduce the heat to low. Add the cheeses and whisk until melted and the sauce is smooth. Remove the pan from the heat and stir in the kale, basil and garlic. Season with salt and pepper.
Make the Pasta: Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until just tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain and set aside.
Assemble the Lasagna: Place an oven rack in the center of the oven. Preheat the oven to 400°. Butter a 9-by-13-inch glass baking dish. Spread 1 cup of the marinara sauce in the bottom of the prepared baking dish. Lay 3 noodles over the marinara. Spread 1/3 of the Filling Mixture evenly over the noodles. Sprinkle with a 1/3 of the shredded short ribs. Repeat with the remaining noodles and filling, making 3 layers of filling and ending with pasta. Spoon the remaining marinara sauce on top and sprinkle with Parmesan cheese. Drizzle with olive oil and bake until the lasagna is heated through and the cheese is beginning to brown, about 25 minutes. Allow to cool for 20 minutes before serving.