Preheat oven to 350°F. Season short ribs generously with salt, pepper, and rosemary.
In a large Dutch oven over medium-high heat, add a drizzle of oil. Sear short ribs on all sides, 5-6 minutes. Remove and set aside.
Add onion, carrots, and celery to the same pot. Cook 5-7 minutes until softened. Stir in tomato paste and cook until caramelized.
Pour in wine, scraping up browned bits. Add crushed tomatoes and broth. Bring to a simmer.
Return beef to pot, making sure sauce covers the meat. Add thyme, rosemary, bay leaves, Parmesan rind, and chopped garlic.
Cover and transfer to oven. Braise 3-4 hours until fork-tender.
Remove beef, discard herbs, bay leaves, and Parmesan rind. Shred beef and return to pot, mixing with sauce.
Meanwhile, cook lasagna noodles in salted boiling water until just before al dente. Drain and rinse with cold water.
Make béchamel: melt butter in a medium saucepan over medium heat. Whisk in flour and cook 1-2 minutes (don’t brown). Slowly whisk in milk. Cook 5-7 minutes until thickened. Season with salt, pepper, and nutmeg. Stir in Parmesan until melted.
Preheat oven to 375°F. Spread a thin layer of ragù in a 9x13 baking dish. Layer: noodles, béchamel, ragù, shredded cheeses. Repeat for 4-5 layers, ending with cheese on top.
Cover with foil and bake 45 minutes. Remove foil, broil 3-5 minutes until golden and bubbly.
Rest 10-15 minutes before slicing.