Preheat the oven to 375 F. Lightly butter an 11" by 15" lasagna pan.
Using a stove-top smoker and following the manufacturer's instructions, smoke the tomatoes, cut side up and lightly salted, for 20 minutes. Let cool, and peel the skins from the tomatoes. Place the skinned tomatoes in a large bowl and, using an immersion blender, purée the tomatoes. Stir in the canned tomato purée and béchamel and season with salt and pepper to taste.
Warm the butter and oil in a large skillet over high heat. Add the mushrooms, salt generously, and cook, stirring occasionally, until the moisture has evaporated and the mushrooms have lightly browned, about 20 minutes. Set aside.
In a medium bowl, combine the ricotta, basil and eggs. Season to taste with salt and pepper.
Spread 1 1/2 cups of the smoked tomato sauce mixture in the bottom of the pan. Layer 1/3 of the noodles, 1 1/2 cups sauce, 1/2 the mushrooms, 1/2 the sausage, 1/2 the ricotta mixture. Repeat the layering, ending with the remaining sauce. Sprinkle the top with the shredded mozzarella.
Cover tightly with foil and bake for 45 minutes. Remove the foil and continue baking until the cheese is golden brown and bubbling, about 30 minutes more. It may be necessary to increase the oven temperature to 450 F to brown the cheese.
Let stand for 10 to 15 minutes before serving.
Serves 12 to 16