Preheat oven to 375°F. Cook pasta as directed; drain well. Lay noodles out in a single layer onto buttered waxed paper to prevent them from sticking together.
Spread 1 cup Béchamel sauce over bottom of a buttered 9x13-inch baking dish. Place 4 noo-dles atop sauce. Spread 1 cup Béchamel sauce over noodles. Spoon 1? cups pesto sauce over the Béchamel sauce. Sprinkle ? cup Parmesan cheese over pesto sauce. Repeat layers 2 more times: 4 more noodles, 1 cup Béchamel sauce, 1? cups pesto sauce, and ? cup cheese. Finish assembly with last 4 noodles, 1 cup Béchamel sauce, and ? cup Parmesan cheese. Cover lasagna with foil and bake for 30 minutes. Uncover and bake 20 more minutes. Let sit 10 minutes before serving.
Melt butter in a large saucepan over medium heat. Add shallots; sauté 2 minutes. Add flour and whisk until smooth. Reduce heat to low and whisk 1 minute. Gradually whisk in milk. Bring to a boil, whisking constantly. Add bay leaf. Reduce heat and simmer until slightly thick-ened, about 5 minutes. Mix in nutmeg. Season with salt and pepper. Remove bay leaf and let cool.
Blend basil, Parmesan, olive oil, pine nuts, and garlic in a food processor until a coarse purée forms. Lightly mix in vegetable broth. Season with salt and pepper. Let cool.
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