In saucepan, cover potatoes with cold water. Bring to boil, reduce heat, cover and simmer until potatoes are tender yet still firm in centre, about 15 minutes. Drain and cool; peel and refrigerate until chilled. Meanwhile, in skillet, melt 2 tbsp (25 mL) of the butter; cook onion and mushrooms until lightly browned. Transfer to bowl; refrigerate until chilled.

Grate potatoes into large bowl; add mushroom mixture, rosemary, and salt and pepper to taste.

In 8-inch (20 cm) nonstick skillet, heat about 1 tbsp (15 mL) of the remaining butter and 1 tbsp (15 mL) of the oil over medium-high heat; press half the potato mixture evenly over skillet. Reduce heat to medium; cook until underside is golden, 5 to 7 minutes; invert onto plate. Heat about 1 tbsp (15 mL) butter in skillet; slide uncooked side of rösti into skillet; cook until golden, 5 to 7 minutes.

Repeat with remaining potato mixture. Garnish with parsley.