First, make the pesto béchamel. Bring a saucepot to low heat, and add the butter. Stir until the butter is completely melted. Then, add in the flour and whisk together until completely incorporated (this is a roux). Add in the salt, pepper, and nutmeg to the roux. Cook the roux for about 3-4 minutes.
Then, reduce the heat to the burner's lowest setting (or take the pot off the heat if your stove runs hot). Add in 1/2 cup of the milk, and whisk together with the roux. The roux will be liquid at first, then will clump up into a dough-y texture. That's what we want!
Continue adding in the milk in 1/2 cup increments, stirring constantly, until the béchamel is thick and creamy. This may take up to 15-20 minutes. Be sure to cook on the lowest heat possible so the sauce doesn't break, and don't walk away from the stove! Note that the béchamel will continue to thicken as it cools.
When it's done, the sauce should be thick enough to coat the back of a spoon if you dip one in there. Now, take the pot off the heat. Then, add in the dijon mustard and pesto and whisk until fully incorporated. When you're finished, you can set the béchamel aside.
Now, preheat your oven to 375 degrees F.
Take your summer squash and slice it into thin slices. I recommend using a mandoline slicer on its medium setting. If you're using a squash that has seeds in the middle, be sure to scoop those out first. Continue doing this until you have 8 cups of sliced squash.
Add the squash to a large bowl, then drizzle with olive oil and sprinkle with salt and pepper. Mix, until all of the slices are coated.
Now it's time to assemble your lasagna! You can use a traditional casserole dish or a large braiser with a lid. Start with a layer of the pesto béchamel about 1/8" thick on the bottom, then add your layers in the following (suggested) order - lasagna noodles, squash, Italian blend cheese, parmesan, and repeat.
Once you're done, be sure the last layer is topped with cheese.
Cover your lasagna with foil or with a lid, then bake at 375 for 40 minutes.
After 40 minutes, uncover the lasagna and bake for another 10-20 minutes, or until the cheese on the top has started to brown. If the cheese is still not brown after 20 minutes, broil the top instead of continuing to bake.
When done, serve the lasagna. Enjoy!