Wash spinach leaves well in three changes of cold water; shake dry. Melt 1 tablespoon butter in large pan over medium heat. Add half the garlic; sauté until light golden, about 2 minutes. Add half the spinach leaves, cover, and cook, stirring occasionally, until wilted, 4 to 5 minutes. Drain spinach in a colander. Repeat with another tablespoon butter and remaining garlic and spinach. When spinach is cool enough to handle, squeeze to rid it of liquid. Roughly chop spinach; place in medium bowl. Add ricotta, 2 teaspoons salt, 1 teaspoon pepper and the 1/2 teaspoon of ground nutmeg.
Melt 2 tablespoons butter in a large skillet over medium heat. Add a third of the mushrooms; season with 1 teaspoon salt and 1/4 teaspoon pepper. Sauté until mushrooms are softened and browned, about 10 minutes. Deglaze skillet by pouring 1/4 cup Madeira into hot skillet with mushrooms and using a wooden spoon to loosen bits cooked onto skillet. Cook mushrooms until liquid has almost evaporated. Transfer cooked mushrooms to medium bowl. Repeat with another 2 tablespoons of butter, another third of the mushrooms, and 1/4 cup Madeira. (Reserve the final third of mushrooms and Madeira for topping.) Add two-thirds of the chopped parsley to bowl with cooked mushrooms; stir.
Add 4 cups of the milk to 2 jars of alfredo sauce reserving last 1/2 cup for the sauce.
Heat oven to 350°. Set aside 1/2 cup sauce. Assemble lasagna: Spread 1/2 cup sauce in bottom of 9-by-13-inch baking pan. Place layer of lasagna sheets in pan, trimming to fit; spread 1 cup spinach mixture, 1 cup mushroom mixture, and 1/2 cup sauce on top of lasagna sheets. Repeat layers several times. For last layer, place a layer of lasagna sheets on top; spread 1/2 cup sauce over lasagna sheets. Sprinkle with 1/2 cup grated cheese. Bake lasagna until top is golden brown, 1 to 1 1/4 hours. Let stand 20 minutes before serving.
Just before serving, melt remaining 2 tablespoons butter in the skillet over medium heat. Add remaining third of the uncooked mushrooms; season with the remaining teaspoon salt and 1/4 teaspoon pepper. Cook until golden and tender, about 10 minutes. Deglaze skillet, with mushrooms in it, with the remaining 1/4 cup Madeira. Stir in the remaining third of fresh parsley.
In a small saucepan, combine reserved 1/2 cup sauce with remaining 1/2 cup milk. Over medium heat, whisk until warm and smooth. Spoon cooked mushrooms over each serving, or serve on the side. Serve lasagna with warmed sauce.