Heat the olive oil in a large saucepan, add the meat and onions and cook over medium heat. Stir for about 10 minutes or until the meat is no longer pink. Season with salt and black pepper.

Add in the mushrooms, thyme, vinegar, tomato paste, tomato sauce and chicken broth. When the meat mixture starts boiling, reduce the heat and cook until most of the liquid has evaporated. Stir regularly.

To prepare the béchamel sauce, melt the butter in a saucepan. Add in the flour and stir over medium heat until the mixture becomes crumbly. Slowly mix in the milk and continue stirring until it is completely absorbed.Season with salt, pepper and nutmeg and mix well. Remove from heat.

Preheat oven to 350 F.

Spread 1 cup of béchamel evenly over the bottom of a baking pan.

Arrange one layer of the lasagna squares over the béchamel. Then spread a layer of the meat mixture over the lasagna and cover with a generous layer of béchamel. Sprinkle with grated cheese. Repeat two more times with lasagna squares, meat mixture and béchamel. The final layer should be covered with béchamel. Sprinkle the final layer with nutmeg and cheese.

Bake in the oven for about 40 minutes or until the top of the lasagna becomes golden. Allow to cool 10 minutes.

Serve warm.