Position an oven rack 5 to 6 inches from the broiler unit, and preheat the broiler. Line 4 rimmed baking sheets with wire racks.
For the lasagna chips: Bring a large pasta pot of salted water to a boil. Fill a large bowl with ice water.
Cook the lasagna noodles 2 minutes shy of the package directions. Drain the noodles and immediately plunge them into the ice water. Drain the noodles, put them on the prepared baking sheets and pat dry with paper towels. Cut the noodles into triangles using kitchen scissors so they resemble tortilla chips.
Put the oil and 2 tablespoon water in a large bowl. Add the lasagna chips and toss to combine. Add the Parmigiano-Reggiano and red pepper flakes. Season with salt and pepper and toss until evenly coated. Put the coated lasagna chips on the prepared baking sheets. Broil in batches until golden and crispy, 8 to 10 minutes. Remove and allow to cool.
For the Bolognese sauce: Heat the olive oil in a Dutch oven or braiser over medium heat. Add the ground beef, season with salt and pepper and cook, breaking up the meat with a wooden spoon, until browned, about 5 minutes. Meanwhile, put the onion, carrots, celery and garlic in a food processor and pulse until finely chopped. Stir the vegetable mixture into the ground beef, season with salt and pepper and cook for 1 minute. Stir in the tomato paste and add the thyme sprigs. To deglaze the Dutch oven, add the wine, scrape up the browned bits with a wooden spoon and cook for 30 seconds before stirring in the tomato puree. Cook over high heat, stirring occasionally, until the liquid reduces and the sauce thickens, about 10 minutes. Remove the Dutch oven from the heat, stir in the milk and season with salt and pepper. Set aside until ready to use.
For the bechamel: Melt the butter in a medium saucepan over medium heat. Add the shallot and thyme leaves and cook until fragrant, about 1 minute. Stir in the flour and cook until the mixture bubbles, about 30 seconds. Slowly whisk in the milk and cook over low heat until thickened, about 2 minutes. Season with salt and pepper. Turn off the heat and stir in the fontina.
For the spicy pesto: Combine the walnuts and garlic in a mortar and pestle and pound to a paste. Add the basil leaves, a handful at a time, grinding down to a paste with each addition. Add the cheese and drizzle in the Calabrian chili oil and olive oil. Season with salt and combine well until emulsified.
To serve, put the lasagna chips on a large serving platter, then top with the Bolognese sauce, Bechamel and Spicy Pesto.