Melt butter in a medium saucepan and continue cooking until the milk solids turn golden brown and develop a nutty aroma. Let cool slightly, then strain through cheesecloth or a coffee filter. Set aside.
Preheat the oven to 375°F. Place one sheet of phyllo on a clean work surface and brush with melted butter. Top with another sheet and repeat until you have 8 layers.
Using a biscuit cutter, lightly mark rounds in the phyllo without cutting completely through. Arrange parsley leaves and caraway seeds decoratively within three of the marked rounds.
Add 2 final sheets of phyllo, brushing each with butter. Cut out the rounds using the biscuit cutter.
Bake for 10 to 12 minutes, until golden brown. Once cooled slightly, carefully remove the top layer from the rounds that will be used inside the Napoleons, reserving the decorated tops for the final layer. Repeat the process as needed to make enough rounds for 4 servings.
Mushroom Mixture
Tear the oyster and chestnut mushroom caps into approximately ½-inch pieces, reserving the stems for the jus. Pull the lion’s mane apart into small shreds.
Heat brown butter in a medium sauté pan. Add the shallot and garlic and cook until translucent and lightly caramelized.
Add the mushrooms and additional brown butter as needed. Cook until just tender.
Deglaze with ¾ cup Marsala wine and cook until the liquid is almost completely reduced. Season with salt and pepper to taste.
Mushroom Jus
In a medium saucepan, caramelize the shallots. Add the garlic, bay leaves, peppercorns, and reserved mushroom stems. Reduce the heat and cook until the garlic has softened without browning.
Deglaze with 1 cup Marsala wine and reduce by half.
Add the water, salt, and pepper. Continue cooking until the liquid has reduced significantly and becomes a rich, dark sauce. Strain before serving.
Roasted Tomato Garnish
Toss the cherry tomatoes with olive oil and kosher salt. Roast until the skins begin to blister. Set aside.
To Serve
Spoon about 1 teaspoon mushroom jus into the center of each plate and place a phyllo round on top.
Add about 1½ spoonfuls of the mushroom mixture, spreading it to the edges of the phyllo. Sprinkle with crumbled goat cheese. Repeat with another phyllo layer, mushroom mixture, and goat cheese.
Finish with a decorative phyllo top. Spoon additional jus around the plate and garnish with roasted cherry tomatoes, additional mushrooms, and goat cheese. Finish with a light drizzle of brown butter.