In a large skillet, stirring frequently, cook butter over medium heat until golden brown and most of the foam has subsided. Immediately transfer butter to a small bowl. Stir in hazelnuts and vinegar and season with salt and pepper. Cover to keep warm. Wipe out pan with a paper towel.
Season scallops generously with salt and pepper. Place same skillet over medium-high heat. When skillet is hot, cook scallops in two batches until browned and opaque in the center. Turn over once, about 2 minutes total.
Top Arugula with scallops and spoon hazelnut butter over the scallops to dress.