To make the egg pasta, place the flour and eggs in the bowl of a stand mixer and mix on medium speed. Mix until the dough comes together. Shape into a ball, cover with plastic wrap, and let rest for about 30 minutes. Alternatively, you can use pre-made lasagna pasta.
Meanwhile, prepare the béchamel sauce and the filling.
For the béchamel sauce, pour the milk into a small saucepan along with the nutmeg, salt, and pepper to taste. Heat until just below boiling. In another saucepan, melt the butter, add the flour, and whisk, cooking the roux for a couple of minutes.
Remove from the heat and add the hot milk a ladle at a time, stirring constantly to prevent lumps. Once the mixture is smooth, pour in the remaining milk. Return the pan to the heat and cook until boiling, stirring constantly. Reduce the heat, cook for another minute, and remove from the heat. Cover with plastic wrap and set aside. If you prefer, you can use ready-made béchamel sauce.
Briefly boil the green beans and diced potatoes in salted water (they will finish cooking in the oven).
Once the dough has rested, roll out the lasagna sheets. Attach the pasta sheeter attachment to the stand mixer and roll out the dough. Blanch the pasta in boiling water and place it on a kitchen towel to drain.
Butter the baking dish (alternatively, you can use four single-serving pans), spread a couple of spoonfuls of béchamel sauce on the bottom and place the egg pasta on top.
Cover again with béchamel sauce and sprinkle with green beans, potatoes, pitted olives, stracchino cheese, and pesto. Cover with the egg pasta and repeat the same steps to form two more layers. Finally, cover with egg pasta, béchamel sauce, pesto, stracchino cheese, and Parmesan.
Cook in a static oven at 200° for approximately 20 minutes plus grill for final browning.
