1 lb. large sea scallops (about 12) muscles removed, cut in half horizontally
1 to 2 bunches (about ½ pound total) arugula (preferably baby) washed well and dried
In a large skillet, stirring frequently, cook butter over medium heat until golden brown and most of the foam has subsided, about 4 minutes. Immediately transfer butter to a small bowl. Stir in hazelnuts and vinegar, and season with salt and pepper. Cover hazelnut butter to keep warm, and set aside. 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 (to avoid steaming), until browned and opaque in center, turning over once with a thin-bladed metal spatula (see note, below), about 2 minutes total.
Divide arugula among four serving plates; top with scallops. Spoon hazelnut butter over scallops.