Combine rice, parmigiano-reggiano, oregano, salt, pepper, and egg. Form rice mixture into eight 2" round cakes. Transfer cakes to a sheet tray and refrigerate for 30 minutes to allow them to firm up.
Put flour on a plate; dredge cakes in flour. Heat oil and 3 tablespoons of the butter in a 10" skillet over medium-high heat. Working in 2 batches, fry cakes, turning once, until golden brown, about 4 minutes per side.
Garnish each cake with a thin slice of softened butter and sprinkle with finely chopped flat-leaf parsley leaves.