Warm 2 tablespoons of oil in a large skillet over medium heat.
Add mushrooms, onion, garlic, salt, pepper, and oregano.
Sauté until the mushrooms release their juices and the liquid evaporates, about 10 minutes and remove from heat.
Transfer the cooked mushrooms to a cutting board and clean the skillet.
Finely chop the mushrooms and place them in a large bowl.
Mix in bread crumbs and oats with the chopped mushrooms.
Season with additional salt and pepper if needed.
Add the Parmigiano-Reggiano cheese and eggs, stirring to combine.
Let the mixture sit for about 15 minutes, allowing the bread crumbs to absorb any excess moisture.
With damp hands, shape the mixture into patties, using about 1/4 cup for each.
Heat the remaining 2 tablespoons of oil in the skillet over medium heat
Fry the patties until they are golden brown and cooked through, about 2 1/2 to 3 minutes per side.