in a small bowl, combine the butter, cilantro, garlic, shallot, 1/4 teaspoon pepper and salt to taste. Using whatever tool you use to zest citrus; zest the lemons into the butter. Stir and set aside. You can cut the lemon into wedges to serve with the fish.
Heat a dry medium skillet over medium-high heat until hot but not smoking. Pat the fish dry with some paper towel, then season generously with salt and pepper. swirl the oil into the hot pan and let it get shimmery hot. Add the fish (if your fish has skin, place it skin side down first). Leaving the fish alone until it releases itself, if you move it before it has a chance to sear and make a crust, it will stick and tear. It will still taste fine but not be as pretty.
Cook until the bottom side is golden and caramelized, 3-4 minutes. Using a thin spatula, being gentle. flip the fish, add the herbed butter to the pan, and cook until the fish is thoroughly cooked thru maybe 3 minutes.
During the final minute or so, using a spoon baste, or drizzle the butter mixture over the fish. Place the fish on your plates, drizzle with some of the pan sauce. Season to taste with salt and pepper, garnish with some cilantro and serves with the lemon wedges.