Start with the pasta. Combine 3 1/2 cup flour, 1/4 cup water, 2 tbsp. of the olive oil, both eggs, and the chopped spinach to form a rough dough.
Knead the dough in the bowl until smooth and then let rest in the fridge for half an hour to firm up.
Heat the remaining 2 tbsp. oil in a large saucepan over medium heat.
Add pancetta and cook until brown.
Add onion, celery, carrots, and cook 5-8 minutes until soft.
Add beef, along with a pinch each of salt, pepper, and nutmeg. Cook until beef is no longer pink
Add wine and cook until it has evaporated. Whisk in tomato paste and 2 cups water (You might omit water and substitute 1 cup milk if you prefer a consistency more like Bolognese when it served as a pasta sauce, instead of in lasagna.)
Bring the liquid to a boil, then reduce to a simmer and cook uncovered for 30 minutes.
While the meat sauce cooks, bring a large pot of water to a boil.
As water heats up, remove the pasta from the fridge and turn it out on a lightly floured surface. Divide into three sections, rolling out each into 1/8-inch thick sheets. (Alternately, use a pasta maker.)
Cut the sheets of pasta into wide rectangular ribbons that fit your baking pan. Gather scraps and re-roll to yield additional ribbons.
Let the pasta rest while you start the béchamel. Melt 6 tbsp. of butter in a saucepan over medium heat.
Whisk in remaining cup of flour and cook 3-4 minutes until lightly golden.
Whisk in hot milk in a slow stream until smooth, finishing with a dash each of salt and nutmeg.
Barely simmer the béchamel on low for 10 minutes, stirring occasionally.
Drop the lasagna ribbons into the boiling water and cook for one minute before draining and rinsing them.
Preheat oven to 350 degrees, and butter a baking dish.
When the sauces are finished and the pasta ready, layer 1/3 of the pasta on the bottom of the dish. Top with 1/3 of the bechamel, 1/3 of the Bolognese, and 1/3 of the parmesan. Repeat twice more. Dot the top with the remaining 2 tbsp. butter.
Bake 40 minutes, till the liquids are bubbling hot
Remove from oven and let rest 10 minutes before serving.