In large frying pan, heat butter over medium high heat and saute the onion until soft and golden brown while stirring occasionally, about 5-10 minutes. Remove from heat and allow onion to cool.
In large bowl, combine cooled cooked onions with salmon, egg, mayo, bread crumbs, lemon zest, lemon juice, basil, and salt. Combine well and form into 4 patties.
Coat both sides of each salmon patty with grated parmesan cheese.
Using same pan you cooked the onions in, heat the olive oil over medium high heat. When the oil is hot, add the salmon patties and cook until golden brown, flipping only once, about 5-10 minutes on each side.
Serve hot with your favorite sides or serve cold on a salad.