Pre-heat oven to 375 F.
Line a baking sheet with parchment paper. Brush mushroom caps lightly with olive oil. Place caps gill side up on parchment and sprinkle salt and freshly ground pepper over them to taste.
Roast caps in preheated oven for 15 minutes.
Add 1 teaspoon olive oil to a medium skillet over medium heat. Add garlic and cook briefly. Add arugula and cook until wilted. Add salt to taste.
Remove caps from the oven. Divide the cooked arugula and place an even amount in each mushroom cap. Top with 1 ounce of mozzarella and a handful of the chopped tomatoes. Sprinkle dried oregano over tomatoes and top with Parmigiano-Reggiano cheese. Bake for 10- 15 minutes or until cheese is melted.
