Mix 3 tablespoons butter, the lemon zest, juice, shallots and 2 tablespoons chives until well combined. Place the butter onto a small piece of plastic wrap and, using both ends of the plastic wrap, spin it around to make a cylinder. Chill for at least 30 minutes.
Using needle nose tweezers, remove all the bones from the trout. Melt the remaining 2 tablespoons butter over moderate heat in a large nonstick skillet. Sprinkle the trout with salt and pepper. Place the fish skin-side down and cook until golden brown, 3 to 4 minutes. Flip and cook for another 3 minutes. Using a sharp paring knife, slide the knife into the thickest part of the fish. If you don't feel any resistance, your fish is done. Transfer the fish to serving plates, top with butter and garnish with the remaining chives.
BYOC: The beauty of this butter is you can make it anything you want. Why not try limes and chiles, lemon and mint, rosemary, thyme and sage? Anything goes when you are in the kitchen.