Seared Scallops with Brown Butter and Lemon Pan Sauce

Cut 2 lemons in half and squeeze juice into a measuring glass or small bowl; you should have 1/4 cup juice. Set aside. Using a paring knife, cut ends off remaining lemon to expose flesh. Upend lemon on a cut end and remove peel and white pith from lemons; discard. Cut between membranes to release segments into bowl with juice; squeeze membranes to get any last drops of juice. Fish out any seeds; set aside. Thinly slice chives and place in a small bowl; set aside.
Pull side muscle off scallops, if needed; pat dry. Season lightly on both sides with salt and pepper. Heat a large skillet, preferably stainless steel, over medium-high. Pour in oil to lightly coat surface (2–3 Tbsp.); heat until it shimmers and you see first wisps of smoke. Swiftly place scallops into skillet, flat side down, and cook without touching, tossing, or fussing until underside is deep golden brown, 3–4 minutes. Use a thin spatula or tongs to gently turn over; if they resist, cook another 30 seconds and try again. Cook on second side until flesh at top and bottom looks opaque but there is still a faintly translucent strip in the middle, 1–2 minutes, depending on size. Transfer scallops to a plate.
Pour off any oil in skillet and set over medium heat. Add butter and cook, swirling, until butter foams, then browns, about 2 minutes. Add reserved lemon juice and segments; energetically stir and swirl pan to emulsify sauce. Mix in capers and reserved chives and spoon pan sauce around and over scallops.