Preheat oven to 400 degrees F with rack in middle position.
In a large bowl, mix together cream cheese, mayonnaise, Parmigiano Reggiano, Italian 4-cheese shred, minced garlic, and garlic salt with a wooden spoon or hand mixer until fully combined. Stir in prepared spinach and artichoke hearts by hand.
Clean mushrooms and remove and discard stems. Fill mushrooms with spinach-artichoke mixture by the heaping tablespoonful, pressing filling into the mushroom cavity and slightly mounding.
If using shredded mozzarella topping, sprinkle evenly over the filled mushrooms.
If using Panko topping, stir together breadcrumbs, Parmigiano Reggiano, and salt in a small bowl. Add 1 tablespoon olive oil and toss until combined. Sprinkle the breadcrumbs over the filled mushrooms, pressing gently to adhere.
Grease a large baking sheet or oven-safe skillet with about 2 tablespoons olive oil. Arrange filled and topped mushrooms and bake 23-30 minutes until mushrooms are cooked, filling is hot, and topping is lightly browned. If using breadcrumbs, check about halfway through to make sure they aren't browning too quickly. If they are, gently lay a piece of foil over the mushrooms and continue cooking. Serve hot.
