Prepare barbecue for medium-high heat. Cook butter within small skillet until brown. Stir the butter often, for about 4 minutes. Remove from the heat. Mix oil, shallots, capers, vinegar, rosemary and season with vinaigrette and taste with salt and pepper.
Mix white beans, lemon peel, and vinaigrette in medium skillet. Season the fish with salt and pepper.
Brush little vinaigrette over the flesh side from trout, sprinkle with salt and pepper. Place skillet with white beans in edge of grill to warm it enough. Grill trout and flesh side down until it gets brown. Usually it takes about 3 minutes. Turn out over trout, grill until it cooked well for next 2 minutes. Transfer trout to plates and mix arugula into warm beans, spoons to trout.