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. Remove from the heat.

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 and 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.

Heat remaining 2 tablespoons oil in the skillet over medium heat. Pan-fry patties in the hot skillet until golden brown and cooked through, 2 1/2 to 3 minutes per side.