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Begin by rinsing the scallops gently under cold water. Thoroughly pat the scallops dry using paper towels. Ensuring they are dry is crucial for achieving a beautiful sear, as surface moisture inhibits browning.
Place a 10- or 12-inch nonstick skillet over medium-high heat. Allow it to heat for 1 to 2 minutes. Add the olive oil and 1 tablespoon of butter to the hot skillet, and wait until the butter is melted and the mixture is hot. Pat the scallops dry once more, and carefully arrange them in the pan in a single layer, ensuring not to overcrowd. Season generously with kosher salt and freshly ground black pepper. Let the scallops sear undisturbed until a golden-brown crust forms on one side, about 2 to 4 minutes.
Using tongs, carefully flip the scallops and sear the other side until it is equally browned and the scallops feel almost firm to the touch, another 2 to 4 minutes. Remove the pan from the heat and transfer the seared scallops to a plate. Set them aside to rest briefly. Allow the pan to cool slightly for about a minute before proceeding with the sauce.
Return the skillet to medium heat. Add half of the remaining butter (1 1/2 tablespoons) to the pan, along with the finely chopped leeks. Sauté the leeks until they begin to soften, approximately 1 minute. Pour in the dry white wine and allow it to simmer until it has reduced by about half, which should take another 1 to 2 minutes.
Stir in the chopped fresh parsley, chives, and grated lemon zest. Reduce the heat to low. Add the remaining butter and whisk continuously until the butter is fully melted and emulsified into the sauce, creating a luscious herb butter sauce.
Return the seared scallops, along with any accumulated juices from the plate, to the pan with the herb butter sauce. Gently roll the scallops in the sauce to warm them through and coat them evenly. Taste the sauce and adjust the seasoning with salt and pepper as needed.
Serve the pan-seared scallops immediately, accompanied by lemon wedges on the side for squeezing over the scallops. Enjoy!