Bring stock to the boil in a small pan. Reduce heat; cover and simmer slowly until needed. Heat oil and butter heavy-based pan. Add onion; stir over medium heat 3 minutes or until softened. Add rice; cook further 2 minutes. Add mushrooms and cook for 3 minutes or until soft.
Add hot stock 1/2 a cup at a time, stirring constantly, until all stock is absorbed. Repeat process until all stock has been added and rice is just tender and creamy, stirring constantly. (This will take about 20 minutes.) Stir in cheese and remove from the heat.
Transfer the mixture to a bowl to cool; refrigerate for at least 1 hour. With wetted hands, shape 1/4 cupfuls of mixture into flat rounds. Refrigerate for 15minutes.
Heat about 2cm oil in a non-stick pan. Cook fritters 3-4 minutes each side, until golden and crisp. Drain on paper towels. Served with relish if desired.