Preheat your oven to 350 degrees.
In a large sauce pan, melt the butter over medium heat. Add the garlic and cook until fragrant, about 30 seconds. Be careful not to burn it. Add the flour and whisk into the butter until it is completely combined. Allow it to cook and bubble.
Very slowly add the milk. Add it about a tablespoon at a time, whisking the milk into the butter/flour mixture until absorbed before adding more. Continue adding slowly until all the milk is whisked in. Bring to a low boil and then turn down the heat.
Stir in the cheese, first the parmesan, allow it to melt, then add the cheddar one cup at a time allowing to melt before adding the second cup. Stir in the salt and pepper and remove from the heat.
Spread a few spoonfuls of the sauce over the bottom of a 9 by 13 inch pan so that it is completely coated. Then add three lasagna noodles. Cover with 1/3 of the artichokes, 1/3 of the spinach, 1 cup of chicken and 2/3 cup of cheddar. Add a few spoonfuls of the white sauce. Repeat two more times.
Add three more lasagna noodles. Top with the remainder of the white sauce and the remainder of the cheese.
Bake for 35 minutes, uncovered. Broil on low for about five minutes, keeping a close eye on it so it does not burn but browns nicely.
Let sit for 10 minutes after removing from the oven and then serve.