Preheat the oven to 400°F and line a rimmed baking sheet with parchment paper.
Remove the stems from the mushrooms and finely chop them, measuring about 1 1/2 cups. Using a spoon, gently scrape out and discard the gills from the mushroom caps to make space for the filling.
Place the mushrooms on the prepared baking sheet, drizzle with 2 tablespoons of the olive oil, then season with 1/4 teaspoon kosher salt and 1/8 teaspoon black pepper.
Gently toss by hand to coat evenly. Arrange the mushrooms with the stems facing up.
In a large skillet, heat the remaining 1 tablespoon olive oil over medium-high heat until shimmering.
Add the chopped mushroom stems and the finely chopped shallot. Cook, stirring occasionally, until tender and most of the moisture has evaporated, about 3 to 4 minutes.
Stir in the minced garlic, remaining 1/2 teaspoon kosher salt, and 1/8 teaspoon black pepper.
Cook for about 30 seconds, until fragrant.
Transfer this mushroom mixture to a medium bowl.
Stir in the panko breadcrumbs, chopped parsley, shredded mozzarella, grated Parmesan, and lemon zest if using.
Divide the filling evenly among the mushroom caps.
Bake in the preheated oven until the mushrooms are tender, the filling is heated through, and the tops are golden brown, approximately 16 to 18 minutes.
Garnish with additional chopped fresh parsley if desired before serving.