Season beef medallions with salt and pepper; set aside.

Heat vegetable oil in a skillet over high heat. Cook beef medallions until browned, about 2 minutes per side; transfer to a warm plate.

Reduce heat to medium; add mushrooms, 1 tablespoon unsalted butter, olive oil, and pinch of salt to skillet. Cook and stir until mushrooms are very well browned, about 10 minutes.

Stir in tomato sauce; cook and stir until thickens, about 5 minutes.

Add Marsala; increase heat to high. Cook and scrape browned bits from bottom of skillet until most of Marsala evaporated. Stir in broth and bring to a boil; cook and stir until reduced by half, about 5 minutes.

Reduce heat to low; add beef medallions back to skillet and simmer until cooked to medium-rare, 2 to 3 minutes. Transfer beef medallions to serving plates.

Off heat, stir remaining 1 1/2 tablespoons butter and oregano into sauce until butter melts. Taste for seasoning; adjust if necessary.

Spoon sauce over beef medallions and serve immediately.