Preheat oven to 400°F. Line baking sheet with parchment paper.
Prepare mushrooms: Wipe clean with damp paper towel. Remove stems and set aside. Brush caps inside and out with olive oil. Season with salt and pepper. Place hollow side up on baking sheet.
Prep stems: Finely chop reserved stems. Place in kitchen towel and squeeze out as much moisture as possible.
Sauté aromatics: Melt 2 tablespoons butter in small skillet over medium heat. Cook shallots 2-3 minutes until softened. Add garlic and cook 30 seconds. Add chopped stems and cook 3-4 minutes until dry. Season with salt and pepper. Cool slightly.
Make filling: Beat cream cheese in medium bowl until smooth. Add cooled mushroom mixture, mozzarella, Parmesan, panko, parsley, thyme, oregano, and red pepper flakes. Mix until evenly combined.
Season: Taste filling and adjust seasoning. Should be bold and well-seasoned.
Fill caps: Using small spoon or scoop, generously fill each cap, mounding slightly above rim. Pack firmly.
Make topping: Combine remaining panko, Parmesan, and melted butter in small bowl. Stir until breadcrumbs are evenly coated.
Add topping: Sprinkle breadcrumb mixture evenly over each filled mushroom. Press gently to adhere.
Bake: Bake 20-25 minutes until mushrooms are tender, filling is hot and bubbly, and topping is golden brown.
Broil if needed: Broil 1-2 minutes for extra golden color, watching carefully.
Rest: Let rest 3-5 minutes before serving.
Garnish: Transfer to platter, sprinkle with fresh parsley, drizzle with any pan juices. Serve immediately.