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In a medium saucepan, melt the butter over medium-high heat. Add the shallots and cook, stirring frequently, until translucent, about 3 minutes. Add in the garlic and cook for another minute. Stir in the flour and cook, stirring frequently and controlling the heat to prevent browning, about 1 minute. Slowly whisk in the milk and chicken broth. Bring the mixture to a boil and continue whisking until the sauce has thickened and coats the back of a spoon, about 10 minutes. Remove the saucepan from the heat and stir in 1/4 cup of the prepared pesto. Season with salt and pepper.
Whisk 1/4 cup of the pesto cream sauce and the remaining 2 tablespoons prepared pesto into the ricotta mixture. Set aside the remaining sauce.
Preheat the oven to 350 degrees F. Bring a large pot of water to a boil and add salt.
Working in batches, add the pasta sheets to the boiling water and cook for 2 minutes; transfer to a wire rack set over a rimmed baking sheet.
Grease a 9- by 13-inch baking dish with the olive oil. Add 1/2 cup of the pesto cream sauce to the bottom of the baking dish. Place 1 pasta sheet on top of the sauce. Spread 1/3 cup of the ricotta mixture evenly over the pasta and sprinkle with 1/4 cup of the mozzarella mixture. Drizzle 1/3 cup pesto cream sauce over the cheese. Repeat the process, layering pasta, ricotta, mozzarella and pesto cream sauce twice more (3 layers in total). Finally finish with the fourth pasta sheet on top, and cover completely with the remaining pesto cream sauce and ricotta mixture. Sprinkle with the remaining mozzarella mixture.
Coat a large piece of aluminum foil with nonstick cooking spray and cover the lasagna; bake until bubbling, about 30 minutes. Remove the foil and turn the oven to broil. Broil until the top is a spotty golden brown, about 3 minutes.