Preheat the oven to 375 degrees F (190 degrees C).
Bring milk to a simmer in a saucepan over medium high heat.
Meanwhile, melt butter in a separate saucepan over medium heat. Add flour; cook and stir for 2 to 3 minutes. Whisk hot milk vigorously into butter flour mixture and bring to a simmer; cook, stirring occasionally, for 5 minutes. Remove from heat. Stir 3/4 cup Parmesan cheese, nutmeg, salt, and pepper into bechamel until smooth.
Pour 1/4 of the meat sauce and 1/4 of the béchamel into the bottom of a 9x13 baking dish; sprinkle with 2 tablespoons basil. Cover with 3 noodles and repeat for 3 layers. Spread remaining meat sauce and béchamel on top; sprinkle with remaining 2 tablespoons basil and remaining 1/4 cup Parmesan cheese. Cover the dish with aluminum foil.
Bake in the preheated oven until edges are bubbling and slightly brown, about 20 minutes. Let rest for 5 minutes before serving.
Oven-Ready Lasagna with Meat Sauce and Béchamel · Dinner Match Lab | Dinner Match Lab