Bechamel/Veggie Lasagna: where to add the Parm?

So I am gonna be sautéing veggies/aromatics and then mixing them into a Bechamel to top each noodle layer of my Lasagna. I am curious if it would be better to add the Parm to my Bechamel, or just sprinkle it on top of the Moz layer (which comes next)?

I add Parm or Mizithra to my bechamel when I make pastitsio or moussaka. I would add it to the bechamel for lasagna, too.

If I wanted separate cheese in another layer, I would use mozzarella. Parm in the bechamel and mozzarella in another layer.