How to Make It
In a small skillet, toast the pine nuts over moderate heat until lightly golden, about 4 minutes; let cool. In a food processor, pulse the basil with the pine nuts and garlic until finely chopped. With the machine on, drizzle in the olive oil and process until a paste forms. Season the pesto with salt and pepper.
In a medium saucepan, melt the 1 1/2 sticks of butter over moderate heat. Whisk in the flour to make a paste and cook until bubbling, about 3 minutes. Gradually whisk in the milk until smooth and bring to a boil. Simmer over moderately low heat, whisking, until the sauce is thick and no floury taste remains, about 7 minutes. Stir in the pesto and nutmeg and season with salt and pepper. Cover and remove from the heat.
Preheat the oven to 400° and butter a 9-by-13-inch ceramic baking dish. In a large pot of salted boiling water, cook the lasagna noodles until al dente, 6 to 7 minutes. Drain the noodles and transfer to a baking sheet.
Spread 1/3 cup of the pesto sauce in the prepared baking dish. Arrange a single layer of the noodles over the sauce. Top with one-fourth of the remaining sauce and one-third of the cheddar. Repeat this layering twice, then top with a final layer of noodles and the remaining sauce. Sprinkle the Parmigiano on top.
Cover the lasagna with aluminum foil and bake for about 30 minutes, until heated through. Uncover and bake for 15 to 20 minutes longer, until the top is golden in spots. Cover loosely and let rest for 15 minutes before serving.