Soak the mushrooms In a small bowl, soak the porcini in 1-1/2 cups warm water until softened, about 10 minutes. Drain, reserving 1 cup of the soaking liquid. Coarsely chop the mushrooms and set aside in a bowl.
Braise the ribs Season the short ribs all over with 1 tsp. salt and 1/4 tsp. pepper. In a 7- to 8-quart Dutch oven, heat 1-1/2 Tbs. of the oil over medium high heat until shimmering. Add the short ribs, in batches if necessary, and brown all over, 2 to 3 minutes per side. Transfer to a plate. Pour off and discard all but a thin layer of fat.
Reduce the heat to medium and add the remaining 1 Tbs. oil to the pan. Add the onion, carrot, celery, and a pinch of salt and cook until the vegetables are tender, 4 to 5 minutes. Add the tomato paste and bay leaf and cook, stirring, until the tomato paste darkens, 1 to 2 minutes. Add the red wine and simmer, stirring and scraping up any browned bits with a wooden spoon, until almost evaporated, 2 to 3 minutes. Add the reserved mushroom liquid and the tomatoes and bring to a simmer. Add the short ribs and any accumulated juices and the porcini; cover, reduce the heat to maintain a gentle simmer, and cook, turning the ribs every 40 minutes, until the meat is fall-off-the-bone tender, 2-1/2 to 3 hours.
Put the ribs on a plate to cool. Pour the sauce into a heatproof bowl and discard the bay leaf. When the fat rises to the surface, skim it off and discard. Season to taste with salt and pepper. When the meat is cool enough to handle, remove and discard the bones and any fat or cartilage from the meat. Shred the meat and set aside in a medium bowl.
Make a béchamel sauce Melt the butter in a 3-quart saucepan over medium heat. Stir in the flour with a wooden spoon and cook until golden, about 2 minutes. Gradually whisk in the milk and cook at a gentle simmer, stirring often, until the mixture thickens, about 10 minutes. Season with 1 tsp. salt. Mix the béchamel sauce into the short rib sauce.
Assemble and bake the lasagne Position a rack in the center of the oven and heat the oven to 350°F.
Spread 3/4 cup of the sauce on the bottom of a 9x13x3-inch baking dish. Cover the sauce with a slightly overlapping layer of cooked noodles, cutting them as needed to fill any gaps. Spread 3/4 cup of the sauce evenly over the noodles. Add 1/2 cup of the shredded meat in an even layer and then sprinkle with 2 Tbs. of the Parmigiano. Add another layer of noodles and repeat the layers as instructed above, to make a total of 4 layers of sauce and meat and 5 layers of pasta—at this point, you should have used all the meat but still have about 3/4 cup of sauce and 1/4 cup of Parmigiano left. Spread the remaining sauce evenly over the top and then sprinkle with the remaining Parmigiano.
Cover the baking dish with foil and bake for 40 minutes. Remove the foil and bake until the top is browned and bubbly, 15 to 20 minutes. Cool for at least 10 minutes before serving. Serve sprinkled with additional Parmigiano, if you like.
