Preheat the oven to 375 degrees F.
Bring a large pot of water to a boil over medium-high heat.
Add the lasagna noodles, 1 tablespoon of the olive oil, and the salt and cook until al dente, according to the package directions.
Rinse the noodles under cold water to stop the cooking process.
Return the noodles to the pot.
In a large saucepan, heat the remaining oil over medium-high heat until hot, about 2 minutes.
Add the onion and the garlic and sauté until fragrant, about 2 minutes.
Stir in the chicken broth and the flour and whisk until smooth and uniform. Don't worry if the sauce is a little lumpy at first, it will smooth out as it simmers and as whisking continues.
Whisk in the heavy cream.
Bring the sauce to a simmer over medium heat.
Let the sauce simmer, stirring occasionally, until thickened, about 5-8 minutes. You want a thickness similar to gravy; it should easily coat the back of a spoon.
Transfer the saucepan from the heat and immediately stir in the spinach.
In a medium bowl, combine the ricotta cheese, 1 cup of the mozzarella, the parmesan, the egg, and the parsley, stirring well to combine.
In a 9x13-inch baking pan, evenly spread 1/2 cup of the white sauce.
Place 3 of the noodles evenly over the sauce.
Spread 1/3 of the remaining sauce evenly over the noodle layer.
Sprinkle 1/2 of the chicken over the sauce layer.
Spread 1/4 of the ricotta mixture over the chicken layer.
Add 3 noodles on top of the ricotta mixture.
Spread 1/2 of the remaining ricotta mixture over the noodle layer.
Add 3 of the lasagna noodles in an even layer to the top of the ricotta mixture.
Spread another 1/3 of the sauce over the noodle layer.
Sprinkle the remaining chicken over the sauce layer.
Spread the remaining ricotta mixture over the chicken layer.
Add the remaining noodles, the remaining sauce, and the remaining mozzarella to the baking dish, in that order.
Lightly grease a large piece of foil, then cover the pan with the greased-side facing the lasagna.
Bake the lasagna for 45 minutes.
Remove the foil and bake, uncovered, until the cheese is melted, golden-brown, and bubbly.
Optionally, turn the broiler on high and broil until crispy on top, about 3-4 minutes.
Let the lasagna stand for 15 minutes.
Slice and serve.