Serves eight.
Look for Athens brand phyllo twin packs (two 8-ounce packs in one box). The phyllo sheets are the perfect size for this dish. The components can be made a day ahead and refrigerated. Wait to assemble the baklava until you're ready to bake it.
FOR THE MUSHROOM LAYER:
3 tablespoons unsalted butter
2 tablespoons finely minced shallot
½ cup finely diced sweet onion
4 cups roughly chopped assorted fresh mushrooms (about 1 pound)
1 tablespoon chopped fresh rosemary
½ cup low-salt vegetable or thicken broth
¼ cup dry white wine (optional)
Kosher salt and freshly ground black pepper
FOR THE RED PEPPER LAYER:
3 tablespoons extra-virgin olive oil
3 large red bell peppers, cut into1/4-inch dice (about 3 cups)
1 clove garlic, finely minced
1 cup finely diced red onion (about 1/2 medium red onion)
1 tablespoon chopped fresh rosemary
Kosher salt and freshly ground black pepper
FOR THE GOAT CHEESE LAYER:
20 to 22 ounces good-quality goat cheese
2 large eggs
2 tablespoons heavy cream
1 tablespoon finely minced rosemary
1 teaspoon freshly ground black pepper
½ teaspoon kosher salt
FOR THE BALSAMIC ROSEMARY SYRUP:
2 cups balsamic vinegar
2 to 3-inch-long sprig fresh rosemary; more sprigs for garnish
1 pound 9x14-inch frozen phyllo dough, thawed in the refrigerator overnight
½ pound unsalted butter, melted; more as needed
Position a rack in the center of the oven and heat the oven to 375°F.
Make the mushroom layer: Melt the butter in a large sauté pan over medium heat. Add the shallot and onion; cook until softened, 3 to 5 minutes. Add the mushrooms, rosemary, broth, and wine if using; season with salt and pepper to taste. Cook until the mushrooms are tender and most of the liquid has evaporated, 8 to 10 minutes. Let cool.
Make the red pepper layer: Heat the oil in a large sauté pan over medium-high heat. Add the peppers, garlic, onion, and rosemary. Season with salt and pepper to taste. Cook, stirring often, until the peppers and onions are well softened, 7 to 10 minutes. Don't let them brown; add more oil if the pan is dry. Let cool.
Make the goat cheese layer: In a food processor or with an electric mixer, blend the goat cheese, eggs, cream, rosemary, pepper, and salt until smooth. Set aside.
Make the balsamic-rosemary syrup: In a medium saucepan, bring the vinegar and rosemary to a boil. Reduce the heat to a simmer and cook until reduced to a thin syrupy consistency, about 15 minutes. Watch closely during the last few minutes of cooking. (The reduction goes fast at the end and if you overreduce it, it may be too thick or even burn. It will thicken more as it cools.) You should have about 1/2 cup syrup. Remove from the heat; discard the rosemary.
To assemble: Unroll the phyllo; keep it covered with plastic wrap and a damp cloth. Liberally brush the bottom of a 9x13-inch baking dish with some of the melted butter. Lay a sheet of phyllo in the dish and, starting from the edges, brush lightly with butter. Repeat with seven more sheets, brushing each with butter.
With a rubber spatula, spread the mushrooms evenly over the phyllo. Repeat the layering and buttering with eight more sheets of phyllo. Spread the goat cheese evenly over the phyllo. Repeat the layering and buttering with eight more sheets of phyllo. Spread the red peppers evenly over the phyllo. Top with eight more phyllo sheets, buttering as you go. Butter the top two sheets as evenly as possible.
With a sharp knife, cut the baklava lengthwise into thirds, cutting all the way to the bottom of the pan, holding the top layers of phyllo in place with your fingers. Next, cut diagonally from one corner to an-other. Continue making diagonal cuts about 2 inches apart to create 2x3-inch diamonds. Bake until deep golden brown, 40 to 50 minutes. Let rest for at least 15 to 20 minutes.
To serve: Drizzle a little of the syrup on each plate. Arrange one or two pieces of the baklava on each plate, drizzle with a bit more syrup, and garnish with a rosemary sprig.