Saute garlic and onions, then add ground beef. Brown the beef. Add carrot and celery, and saute for about one minute. Add the liquids and the rest of the seasonings and let simmer about 10 minutes.
Melt butter in a pot then whisk in flour.Immediately add milk then bring to a boil. Whisk until mixture is smooth and thickens. Once at desired consistency, stir in salt and pepper, then remove from heat.
Start with a a thin layer of bolognese sauce, then add lasagna sheets, followed by more bolognese and then 1/3 of the béchamel sauce. Repeat two more times. There should be a total of three sets of layers.
Top with mozzarella. Place a cookie sheet under the casserole dish to catch any potential drippings. Bake for 45 minutes, then (optional) broil for 2 minutes to brown the top.