To a plate, add the flour and a generous pinch of salt and black pepper. Mix to combine. Pat the skate filets as dry as possible. One at a time, dredge each wing thoroughly in the seasoned flour, making sure the crevices get coated, and shake off any excess.
Heat oil in a large cast iron skillet over medium-high heat. Add the skate wings to the pan, dropping the filets away from your body, and making sure there is enough space between them to not crowd the pan. Sear for 2-3 minutes or until golden and no longer sticking to the pan. Carefully flip (use 2 spatulas to prevent the fish from breaking) and repeat until both sides are golden and the fish is cooked through. Remove and set to the side on a plate.
Reduce the heat to medium and add a tablespoon of butter to the same pan. Once melted, add the capers and green peppercorns. Saute for about 1 minute, then add garlic and saute until soft. Add the sherry wine and simmer until reduced by about half. Gently scrape any bits from the bottom of the pan.
Add the lemon juice and then turn off the heat and add the remaining butter, swirling or stirring constantly to create an emulsified sauce. Spoon the sauce generously over the skate wings. Garnish with parsley and finish with a pinch of flaky salt.