For the minced herb butter: Line a baking sheet that fits into the freezer with parchment paper.
Put the butter in a small mixing bowl and add the parsley. Stir to combine with a rubber spatula. Scoop into a piping bag fitted with a star tip. Pipe the butter into rosettes on the prepared baking sheet. Freeze until the butter is set, about 1 hour.
For the salmon: Line a plate with paper towels and place the salmon on it skin-side up. Using a paring knife, score the salmon by making 2 small slits in the skin on each piece, making sure not to cut into the flesh. Pat the skin with paper towels to make sure it is as dry as possible.
Place a large nonstick skillet over high heat. Once the skillet is warm, add the vegetable oil and heat until shimmering. The skillet needs to be very hot to create a crispy sear at the start.
Season the salmon skin with 1/2 teaspoon of the salt, then place the salmon pieces skin-side down in the oil; reduce the heat to medium. Using a fish spatula, apply even pressure to each piece to make sure it stays flat. Cook, without flipping, until the edges are golden brown and the flesh is turning opaque, 7 to 8 minutes. Season the flesh of the salmon with the remaining 1/2 teaspoon salt and flip over to finish cooking; it will take about 2 minutes more for medium (125 degrees F), depending on the thickness of the fish. Do not flip the fish again or cover the skillet.
Plate the salmon crispy skin facing up. Top each piece with a rosette of herb butter. Serve immediately.
Request center-cut salmon to make sure that all 4 pieces are even in thickness. This will ensure more even cooking.
Extra butter rosettes can be stored in a ziptop bag in the freezer.