Preheat the oven to 400°F (200°C) with a rack in the middle position.
In a large bowl, beat the cream cheese and mayonnaise with a wooden spoon or hand mixer until smooth. Stir in 1/4 cup grated Parmigiano Reggiano, 1/4 cup Italian four-cheese blend, minced garlic, and garlic salt until combined.
Fold the cooked, well-drained spinach, chopped artichoke hearts, and chopped fresh basil into the cheese mixture. Season with salt and pepper to taste.
Clean the mushrooms and remove the stems; discard stems. Fill each mushroom cap with a heaping tablespoon of the spinach-artichoke mixture, pressing the filling into the cavity and mounding slightly.
If using the mozzarella topping, sprinkle 1/3 cup shredded mozzarella evenly over the filled mushrooms.
If using the Panko topping, combine 1/3 cup unseasoned Panko, 2 tablespoons grated Parmigiano Reggiano, and a pinch of salt in a small bowl. Add 1 tablespoon olive oil and toss until the crumbs are moistened, then press the crumb mixture over the filled mushrooms.
Grease a large baking sheet or oven-safe skillet with about 2 tablespoons olive oil. Arrange the filled mushrooms on the pan, then bake for 23–30 minutes until the mushrooms are cooked through, the filling is hot, and the topping is lightly browned. If the breadcrumbs brown too quickly, tent loosely with foil halfway through baking.
Remove from oven and serve hot.