Preheat the oven to 350°F.
Drizzle olive oil into a large heavy-based pot over MEDIUM heat. Add the onion and garlic and sauté for 5 minutes. Add the shredded spinach and cook for another 5 minutes.
Remove the onion and spinach and set them aside. In the same pot, melt the butter. Use a whisk to mix the flour into the melted butter to create a roux. Whisk continuously to remove all lumps. As you whisk, slowly pour in the cream and milk. Continue until the sauce starts to thicken, about 10 minutes.
Take the pot off the heat and mix the cheddar into the sauce. It should thicken up nicely.
Scoop out 2 cups of sauce. Add the onion, spinach, and chicken to the remaining sauce. Mix until all is well coated.
In a large 9x13-inch casserole dish, spread out half of the chicken mix into the bottom of the dish. Layer on lasagna sheets. Repeat until you have three layers.
Spread the cheese sauce that you kept aside over the top. Sprinkle the mozzarella over it.
Cover the dish with foil and then place into the oven in the middle to bake for 40 minutes. Remove the foil and bake for another 15 minutes. Cheese should be golden brown.