Use a boning knife to remove the skin from the salmon if still intact. Cut into four even-sized fillets, about 2-inch wide and 6 ounces in weight, if not already portioned.
Thoroughly dry the surface and skin with paper towels.
Right before cooking, season both sides with salt and pepper.
Heat a 12-inch stainless steel or nonstick pan over medium heat until hot. Add the oil, then turn the heat to medium-high. Once the oil begins to shimmer, about 1 minute, carefully add the salmon one at a time, flesh-side down. Using the back of a spatula, immediately press the fish down into the pan for about 10 seconds. Add the remaining fillets to the pan, pressing each one down before adding the next piece. Reduce the heat to medium. Cook, occasionally pressing down on the flesh, until the surface is golden brown, crispy, and easily releases from the pan, about 4 to 5 minutes. The salmon will be about 75 to 80% cooked through.
Using tongs, carefully flip the pieces over. Gently press the surface to make direct contact with the pan, do not move. Cook until the edges are opaque, and the center is slightly translucent, about 1 to 2 minutes. The internal temperature should read 120°F (49°C) for medium-rare or 130°F (54°C) for medium. Turn off the heat.
Transfer to a paper towel-lined plate to drain the excess grease. Do not discard the pan.
Carefully add vegetable stock, lemon zest, lemon juice, white wine, capers, salt, and pepper to the same pan used to cook the salmon. Stir to combine and bring the liquid to a simmer over medium heat. Reduce by half, about 5 to 7 minutes.
Turn the heat to low, vigorously whisk in 2 tablespoons of butter—taste and season with more salt and pepper as desired. Add the salmon back to the pan and drizzle the sauce over top. Garnish with parsley, dill, and lemon wedges if desired.
Storing: Cool completely, store in an airtight container, and refrigerate for up to 3 days.
Oil substitutes: Grapeseed, avocado, or canola oil.