Place a rack in the center of your oven and preheat to 400°F Line a rimmed baking sheet with parchment paper.
With a small spoon, gently scrape away and discard the mushroom gills. In a small bowl whisk together 2 tablespoons of the oil, the balsamic vinegar, ¼ teaspoon salt, and ¼ teaspoon pepper. Arrange the mushrooms on the prepared baking sheet and brush both sides with the mixture. Turn stem-side down. Roast for 15 minutes while you prepare the filling.
Warm the remaining 1 tablespoon olive oil in a large skillet over medium heat. Add the shallot and cook until softened, 2 to 3 minutes.
Reduce the heat to low. Stir in the garlic and rosemary and cook until the garlic is fragrant, about 30 seconds.
Add the spinach by big handfuls, stirring as you go so it cooks down. Continue to stir until it is completely wilted, about 2 minutes.
Turn off the heat. Add the white beans, then crumble the goat cheese over the top. Add the sun-dried tomatoes, 2 tablespoons of the Parmesan, the basil, and the remaining ¼ teaspoon salt and ¼ teaspoon pepper. Stir until the goat cheese melts slightly and everything is nicely combined. Taste and adjust seasoning.
Turn the mushrooms stem-side up. Divide the filling evenly among the caps, mounding it on top. In a small bowl stir together the panko and remaining 2 tablespoons Parmesan. Sprinkle all over the top of the mushrooms. Mist the tops lightly with olive oil oil or nonstick spray.
Return the pan to the oven and bake until the mushrooms are tender and the filling is heated through, about 10 minutes. To brown the top, turn the oven to broil for 2 to 3 minutes (keep an eye on them and don't walk away!). Sprinkle with additional basil and serve warm.
Notes
TO STORE: Transfer leftover stuffed portobello mushrooms to an airtight container and refrigerate for 3 to 4 days.
TO REHEAT: Warm the mushrooms in the microwave or in a 350ºF oven until heated through.
TO FREEZE: Freeze in an airtight container or freezer bag for up to 2 months. Thaw in the refrigerator and drain off any liquid before reheating.