Preheat the oven to 375°F (190°C) and line a baking sheet with parchment paper.
Prepare the mushrooms: Remove the stems from the mushrooms, and use a spoon to scoop out any remaining gills. Set the mushroom caps aside.
Cook the zucchini: Heat olive oil in a skillet over medium heat. Add the onion and garlic, sautéing until fragrant. Add the grated zucchini and cook until softened and most of the moisture evaporates, about 5-7 minutes. Season with salt and pepper.
Make the filling: Remove the skillet from heat. Stir in the goat cheese and herbs, mixing until the filling is creamy and well-combined. If you like, you can add breadcrumbs to the mixture for added texture.
Stuff the mushrooms: Spoon the zucchini and goat cheese mixture into the mushroom caps, pressing gently to fill them completely.
Bake: Place the stuffed mushrooms on the prepared baking sheet and bake for 20-25 minutes, or until the mushrooms are tender and the filling is slightly golden on top.
Serve: Garnish with extra herbs if desired, and serve hot.