Saute the onion in oil and 4 Tbsps butter on medium heat until it turns translucent, add the mushrooms, add a pinch of salt and a grinding of pepper, turn the heat to high and cook uncovered for about 7 minutes, stirring frequently, until the liquid from the mushrooms evaporates. Set aside.
Make the béchamel, Isn't that a great word? Bechamel is easy to make and you can use it in so many recipes so master it. It's really just a basic white sauce. Heat the milk, I do this in the microwave for 2 minutes on high but you can use another sauce pan and heat it to just before it boils, Melt the butter in a sauce pan on low heat and add the flour all at once. Stir with a wooden spoon until blended for about 2 minutes. Don't let the flour brown. Remove from the heat and add the milk a little at a time, stirring constantly. Add the salt. Return the pan to low heat and continue stirring, thickening the mixture to the consistency of heavy cream, You can use a whisk to beat out any lumps that form but in a lasagna, the lumps will dissolve during baking. If a film forms when you stop stirring, just give it another stir before you use it.
Smear the bottom of the lasagna pan with butter and line the pan with a single layer of pasta, spread a thin layer of the mushrooms, you don't need to cover every inch. Spread a thin layer of bechamel and scatter baby spinach leaves over this, sprinkle with grated parmesan, cover with another layer of pasta and repeat, up to six layers, if you wish, but this is rich, and I've done it with only 3 layers when I was short on lasagna noodles. Finish the top layer with mushrooms, bechamel, spinach and cheese, dot with 2 Tbsps butter and place the pan on the top level of the preheated oven for 10-15 minutes until a light crust forms on top. Let it sit for 10 minutes after you take it out of the oven and serve straight from the pan with extra cheese on the side.