Preheat the oven to 400 degrees Fahrenheit. In a wide and shallow dish, mix together the walnuts, rosemary, salt, and black pepper. Set aside. Rub the salmon filets with the olive oil, then place them in the shallow dish with the walnut mixture, skin side up.
Coat all exposed surfaces of the filet, except the skin, with the mixture and press gently to help the coating stick. Place the filets on a baking sheet lined with tin foil, skin side down. Bake for 10 minutes per inch of thickness (for example, my filets were 2 inches thick so I baked them for 20 minutes). Once done the fish should flake easily with a fork.
While the fish is baking, prepare the brown butter and salad. When browning the butter, it is best to use a stainless steel pan so that you can see the color of the butter change.
Heat the butter in a small shallow frying pan over medium heat. Swirl the pan around a bit every couple minutes to help it cook evenly. Over a period of a several minutes, you'll notice the foam at the top of the butter start to change from light yellow to a dark tan. Once it reaches the dark tan stage and the butter looks light brown and golden, smell it. It should smell nutty and similar to toffee. Remove it from the heat and allow it to cool slightly.