Pat fish dry and lightly salt on both sides. Rest filets for 15 to 20 minutes.
With a sharp knife, make 2 to 3 shallow scores on the skin of each filet.
Heat oil in a stainless steel pan over medium heat just until hot. Add filets, skin side down, and cook 4 to 5 minutes. Carefully turn over and cook for 2 minutes more.
Add butter at this point, let it melt, and spoon over the filets.
Cook until an instant read thermometer, inserted near the center, reads 125 to 130 degrees F (52 to 54 degrees C).
Meanwhile, for beurre blanc, heat shallots and wine to a boil in a small pot. Once boiling, reduce heat to medium-high and cook until liquid is nearly evaporated, 5 to 7 minutes. Turn off heat.
Remove the cold butter cubes from the refrigerator and slowly whisk in a few cubes at a time, making sure the mixture is fully emulsified before adding in more. Once all the butter is combined, whisk in lemon juice and dill. Taste; season with salt. Serve fish with a generous amount of beurre blanc sauce.
