Pat salmon filets dry, then season with 1/4-1/2 teaspoon of salt each, depending on the size. Set aside.
Combine brown sugar, dijon, and 1/4 teaspoon salt in a small bowl. Set aside.
Heat a cast iron (or heavily bottomed, oven-proof) skillet over high heat and wait for it to get very hot. Meanwhile, turn your oven on to broil and position the top rack about 6-8 inches from the top. Add butter, then once it has melted, add the fillets skin-side down. Cook until they have cooked (turned white-ish) about halfway through. Brush a thin layer of sauce over the top of the salmon and place in the oven on the top rack for 90 seconds.
Pull the salmon out and brush a second layer of sauce on the top, then stick it back in the oven for another 90 seconds.
Pull the salmon out and brush with a third layer of sauce, then stick it back in for 2 minutes. When you pull it out for the third time it's done!