Preheat oven to 375°. Prepare a 9×13 pan with cooking spray; set aside.
Heat a large skillet over medium heat and add 1 Tbsp. butter and olive oil. Add the mushrooms and sauté for 8-10 minutes, stirring occasionally. Season with salt.
Add shallot, wine, and remaining 2 Tbsp. butter. Cook, stirring occasionally, until skillet is dry, about 5 minutes.
Combine ricotta and cream in a small bowl; season with nutmeg, marjoram leaves, basil, and oregano.
Spread a thin layer of ricotta mixture in a 9×13 prepared pan and top with a a few pasta sheets.
Spread a large spoonful of ricotta mixture over pasta, scatter some mushrooms on top, then a few pieces of burrata. Sprinkle with Parmesan and a few fresh basil leaves. Repeat layering process 3-4 more times; finish with the last of the Parmesan and a grind or two of pepper.
Cover the baking dish with foil and bake the lasagna for 40 minutes. Remove foil and bake uncovered for about 15 more minutes or until the top is golden. Remove from oven and let the lasagna stand for 15 minutes before serving.