Mushroom and Burrata Lasagnette

Name: Mushroom and Burrata Lasagnette
Ingredients: 3 tablespoons unsalted butter, divided; 3 tablespoons olive oil, divided, plus more for brushing; 1½ pounds mixed mushrooms (such as chanterelle, crimini, and maitake), cut into bite-size pieces; Kosher salt, freshly ground pepper; 1 large shallot, finely chopped; ⅓ cup dry white wine; 1 cup ricotta; ¼ cup heavy cream; 6 fresh pasta sheets (about 7x5 inches) or 12 dried lasagna noodles; 8 ounces burrata or fresh mozzarella, torn into large pieces; 1½ ounces Parmesan, finely grated; 6 teaspoons marjoram leaves; ½ bunch Tuscan kale, ribs and stems removed, leaves torn; 1 tablespoon white wine vinegar