Preheat oven to 425°. Heat 1 Tbsp. butter and 1 Tbsp. oil in a large skillet over medium-high. Add half of mushrooms, season with salt and pepper, and cook, stirring occasionally, until browned and starting to crisp, 8–10 minutes. Transfer to a medium bowl. Repeat with 1 Tbsp. butter, 1 Tbsp. oil, and remaining mushrooms. Return all mushrooms to skillet (reserve bowl) and add shallot, wine, and remaining 1 Tbsp. butter. Cook, stirring occasionally, until skillet is dry, about 5 minutes. Return mushrooms to bowl.

Combine ricotta and cream in a small bowl; season with salt and pepper.

Working in batches, cook pasta in a large pot of boiling salted water, stirring occasionally, until just softened, about 30 seconds. (If using dried noodles, cook until al dente.) Transfer noodles to a large rimmed baking sheet as you go, brushing with oil and overlapping as needed.

Spread a thin layer of ricotta mixture in a small baking dish and top with a pasta sheet (if using dried, use 2 noodles side by side). Spread a large spoonful of ricotta mixture over pasta, scatter some mushrooms over, then a few pieces of burrata. Top evenly with some Parmesan and 1 tsp. marjoram leaves. Repeat layering process (starting with noodles and ending with marjoram) 5 more times; finish with the last of the Parmesan and a grind or two of pepper.

Cover lasagnette with foil and bake until warmed through, 10–15 minutes. Remove foil and continue baking until golden brown, 15–20 minutes. Let cool at least 5 minutes.

Meanwhile, toss kale in a large bowl with vinegar and remaining 1 Tbsp. oil to coat and massage leaves just to soften slightly; season with salt and pepper.

Serve lasagnette with kale alongside.

Do Ahead: Lasagnette can be assembled 1 day ahead. Cover and chill.