Juice one lemon for about 1/8 cup juice and set aside. Cut ends off the remaining lemon and remove peel and white pith; cut between membranes to add segments to bowl with juice, set aside. Thinly slice chives and place in a small bowl; set aside.
Season lightly on both sides with salt and pepper. Heat a large skillet over medium-high. Pour in oil to lightly coat surface (2–3 Tbsp.); heat until it’s hot, you’ll see first wisps of smoke. If the scallops you’ve purchased has a side muscle, go ahead and pull that off. It’s always a good idea to pat dry any moisture before placing the fish in hot oil.
Swiftly place scallops into skillet, flat side down, and cook without touching 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, there should be a faintly translucent strip in the middle, 1–2 minutes. Transfer scallops to a plate.
Pour off any extra 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 lemon segments; stir the pan to thicken the sauce. Mix in capers and reserved chives and spoon pan sauce around and over scallops.