Vegetarian Meatballs

1 tablespoon olive oil 1 pound fresh white mushrooms, finely chopped 1 pinch salt 1 tablespoon butter 1/2 cup finely chopped onion 4 cloves garlic, minced 1/2 cup quick-cooking oats 1 ounce very finely shredded Parmigiano-Reggiano cheese 1/2 cup bread crumbs 1/4 cup chopped flat-leaf (Italian) parsley, packed 2 eggs, divided 1 teaspoon salt freshly…