Preheat the oven to 450°F. Line a rimmed roasting sheet pan with aluminum foil for easy cleanup.
Remove and discard the mushroom stems. Wipe away any dirt with a damp paper towel. Scrape the stem side of the mushrooms with a spoon to remove the gills.
In a small bowl, whisk together the olive oil, balsamic vinegar, and garlic powder. Brush the mixture on the outside and inside of the portobello caps.
Place the coated mushroom caps stem-side down on the prepared sheet pan. Roast for 5 minutes. Remove the mushroom caps and discard any excess liquid that has been released during roasting. Place the caps back onto the sheet pan, stem-side up. Lower the oven temperature to 400°F.
In a medium bowl, combine the spinach, Parmesan cheese, mozzarella cheese, salt, and black pepper. Adjust seasoning to taste. Generously fill each of the mushrooms with the spinach mixture.
Return the stuffed mushrooms to the oven for 5-10 minutes, just enough to warm and soften the cheese. Sprinkle with red pepper flakes, if using, and serve immediately.