Preheat the oven to 400 degrees . Fill a large bowl with ice water and stir in the EVOO to help prevent the cooked noodles from shrinking.
In a large pot of boiling, salted water, cook the noodles until al dente; remove and transfer to the ice bath. Cut each noodle crosswise into 5-inch pieces, then transfer to a kitchen towel to drain.
Meanwhile, in a medium saucepan, melt the butter over medium heat. Add the flour and stir until smooth. Cook, stirring, until light golden, about 5 minutes. Working with about 1 cup at a time, whisk the warmed milk into the butter-flour mixture until very smooth. Bring to a boil, whisking constantly. Cook, whisking, until thickened, about10 minutes; remove from the heat. Season with the nutmeg. Transfer to a bowl and let cool. (If you want to make the sauce in advance, cover and refrigerate until ready to use.)
In a medium bowl, stir 2 cups of the sauce together with the pesto and cheese. Grease four 5- or 6-inch gratin dishes, then place 1 layer of noodles on the bottom of each. Top each with 2 tbsp. spinach and 2 tbsp. of the pesto sauce; add another layer of noodles. Continue layering until you have 4 layers of noodles and 4 layers of the spinach and pesto. Finish with a fifth layer of noodles, followed by a thin layer of pesto; sprinkle with the breadcrumbs. Place the dishes on a baking sheet and bake until bubbling and golden-brown on top, 20 to25 minutes. Serve immediately.