Prepare barbecue (medium-high heat). Cook butter in small skillet over medium heat until brown, stirring often, about 4 minutes. Remove from heat. Mix in oil, then shallots, vinegar, capers, and rosemary. Season vinaigrette to taste with salt and pepper.
Mix beans, lemon peel, and 2 tablespoons vinaigrette in medium skillet. Season to taste with salt and pepper.
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Brush a little vinaigrette over flesh side of trout; sprinkle with salt and pepper. Place skillet with beans on edge of grill to warm through. Grill trout, flesh side down, until brown, about 2 minutes. Turn trout over; grill until cooked through, about 2 minutes. Transfer trout to plates. Mix arugula into warm beans; spoon alongside trout. Rewarm remaining vinaigrette; spoon over trout and beans.
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