Heat 2 tablespoons oil in a large skillet over medium heat. Add mushrooms, onion, garlic, salt, pepper, and oregano. Cook and stir until mushrooms have released their juices and the liquid has evaporated, about 10 minutes.

Transfer mushrooms to a cutting board and clean the skillet.

Chop mushrooms into small chunks, then transfer to a large bowl. Mix in bread crumbs, oats, and season with salt and pepper as needed. Stir in Parmigiano-Reggiano cheese, then eggs. Let stand until bread crumbs have absorbed any excess liquid, about 15 minutes. Use moist hands to form mixture, 1/4 cup at a time, into patties.