Make the Bolognese: Heat oil in a large Dutch oven over medium-high and cook, stirring occasionally, until browned all over.
Reduce heat to medium then add onion, celery, carrot, and garlic and cook, stirring occasionally, until softened, 6–8 minutes.
Add tomato paste and cook, stirring constantly, until paste darkens, about 2 minutes. Add tomatoes, broth and milk then bring to a simmer. Cover pot, turn heat down low and leave to cook for 60-75 minutes at least, checking every 15 minutes and adding liquid as needed.
Make Béchamel: Melt butter in a large saucepan over medium heat. Whisk in flour and cook about 2 minutes. Whisk in milk bring to a simmer and cook, whisking occasionally, until béchamel thickens. Add cheese and season then allow to cool completely.
Assemble lasagna: Preheat oven to 160°C. Cook noodles in a pot of boiling water for 4-5 minutes (they will seem to be too firm) and transfer noodles to a large bowl of cold water to cool. Drain and lie flat in a single layer on parchment lined surface.
Oil a 13x9" glass or ceramic baking dish. Layer your lasanga: ragu, noodles, béchamel, and repeat until you have 4-5 layers. Cover top with béchamel making sure it is covered at the edges.
Bake lasagna until bubbling gently around the edges, about 35-40 minutes. (If you want an even crisper top you can broil it for 5 minutes as well).