Pull side muscle off scallops, if needed; pat dry. Season on both sides with salt and black pepper.
In a large skillet, heat olive oil over medium-high until shimmering. Add scallops and cook, undisturbed, until their undersides are golden brown, about 3 minutes. Gently flip and cook on second side until the flesh looks opaque, about 1 to 2 additional minutes. Transfer to a plate.
Using the same skillet, reduce heat to medium. Add butter and cook, swirling, until butter foams, then browns, about 2 minutes.
Add lemon juice and stir to emulsify sauce. Mix in capers and chives. Plate scallops evenly among 4 plates and spoon sauce around and over them.