Brown the sausage: Heat the olive oil in a large heavy pot over medium-high. Cook the sausage, breaking it up, until browned and crispy in spots, about 7 min. Remove with a slotted spoon, leaving the fat in the pot.
Build the base: Lower heat to medium. Cook the onion until soft, 4 min. Add garlic and chili paste, stir 1 min until fragrant.
Bloom paste and vodka: Stir in tomato paste and cook 2 min until slightly darkened. Add vodka, scrape up the browned bits, and simmer until mostly reduced, 2-3 min.
Simmer the sauce: Add tomatoes, cream, salt and pepper. Simmer uncovered 20 min, stirring occasionally, until thickened. Stir the sausage back in with the butter and parmesan cheese, simmer 5 min, then adjust the salt.
Make the béchamel: Melt the butter in a saucepan over medium heat. Whisk in the flour and cook 1-2 min until pale and foaming (don't brown). Slowly pour in the warm milk, whisking constantly. Cook, whisking often, until it coats the back of a spoon, 6-8 min. Stir in nutmeg, salt and pepper.
Preheat the oven to 190°C (375°F).
Layer: Spread a thin layer of vodka sauce in a 9x13 in (23x33 cm) baking dish. Add a layer of lasagna sheets, then a third of the remaining vodka sauce, a third of the béchamel, and a scatter of mozzarella. Repeat twice more. Finish with the last of the béchamel, remaining mozzarella, and the Parmesan.
Bake: Cover loosely with tented foil and bake 30 min. Uncover and bake 15-20 min more, until bubbling and golden.
Rest and serve: Rest 15 min before slicing. Drizzle with chili crisp and scatter with basil.