Heat oven to 350°F Spread oil over the bottom of a medium-size baking dish.
Arrange potatoes in layers in the baking dish, seasoning each layer with some of the salt and pepper. Scatter onions and asparagus pieces over the potatoes and season with the remaining salt and pepper.
Dot with pieces of butter and sprinkle with Parmesan cheese. Cover with foil and bake for 60 to 70 minutes or until potatoes are fork tender.
Tear cheese slices into pieces; place on top and allow to melt.