1. Squeeze as much liquid out of the shredded potatoes as you can, just right over the sink. Put the squeezed potatoes on a sheet tray, spread out in a thin layer. Sprinkle with the flour and toss to coat/mix that in. 
2. Spread one side of each salmon fillet with some of the mayonnaise (this is the glue for the potatoes). Season with salt and pepper. Press the salmon fillets, mayo side down, into the shredded potatoes, making sure the entire surface (of the mayo side) is covered with potatoes. 
3. Add about 2 TBSP olive oil to a large non-stick skillet set over medium heat. Place two salmon fillets at a time in the skillet, potato side down, and allow the potatoes/salmon to cook until the potatoes are golden brown and crunchy; about 6 to 8 minutes. Flip the fillets and cook for about 45 seconds to one minute more – just so the other side is cooked. Transfer to a plate or tray and season with a little salt and pepper. 
4. Repeat with the remaining fish until all of the fillets are cooked. Serve right away with the lemony brown butter sauce (see below).

For the Simple Lemon Brown Butter Pan Sauce:

Place the butter in a small saucepan or skillet set over medium heat; allow it to melt and then continue to cook it until it turns visibly golden brown and smells nutty; about 2 to 3 more minutes after melted. As soon as this happens, pour into a small bowl. 

Add the lemon juice and season with salt and pepper to taste. This will stay liquid for about half an hour, but if it begins to solidify on you, simply reheat it in the microwave or in the pan on the stovetop over low heat.