It is basically a casserole, Parmentier is always referred to potatoes. As simple as ground beef, with mushed potatoes, baked with some shredded cheese.
100g ground beef
1/2 onion, garlic, chopped
potatoes (boiled then put some butter, cream, muscat and make your mushed potatoes)
1. mix meat, egg, parsley, salt pepper.
2. in butter brown onion and garlic, then add meat, cook.
3. in a casserole add meat at bottom, potatoes on top, then sprinkle with cheese on top.
bake for 20 minutes at 350F or 180C