Using a stove to prepare a skin-on salmon fillet provides benefits that you can't get when using an oven, such as a crispy texture. Although salmon skin crisps in the oven, the direct contact with the saute pan, in combination with a small slit in the skin, keeps the skin flat and helps caramelize its entire surface. This direct contact with the pan creates a desirable browning effect, known as a Maillard reaction, which deepens the overall taste of the fish.
Melt 1 tbsp. of butter in a saute pan over medium-high heat and swirl in 1 tbsp. of olive oil. The olive oil raises the smoking point of the butter without masking its flavor. Cut a 1-inch slit in the skin of the salmon without cutting into the flesh. This prevents the skin from misshaping the fillet as it shrinks from the heat.
Pat the salmon dry with paper towels and season both sides to taste with kosher salt and freshly ground black pepper. You can also use herbs that augment the flavor of salmon, such as dill, fennel and thyme. Place the salmon skin-side down in the saute pan.
Sear the salmon until the skin crisps, which will take approximately three minutes. Turn the salmon over and lower the heat to medium.
Cook the salmon over medium heat until it is cooked through, which will take approximately four minutes. Remove the salmon from the pan, cover it loosely with aluminum foil and allow it to rest for three minutes before serving.