Make the lemon butter sauce by melting the butter in a small saucepan over low heat. Add the minced garlic, lemon zest, parsley and the remaining 1/4 teaspoon salt. Stir to combine and heat for an additional 1-2 minutes, then remove from the heat and set aside. Butter will harden if it gets too cool, simply reheat until melted.
Using kitchen shears, butterfly the lobster tails. Gently loosen the flesh from the shell. Devein the top of the tail, if needed. Skewer, lengthwise, with a soaked wooden skewer or a metal skewer.
Brush the meat with the olive oil and then sprinkle with 1/4 teaspoon of the coarse kosher salt. Set aside.
Heat grill to high medium-high heat. Place lobster tails on the top grill rack, meat side down, for 5-6 minutes. Turn and continue to cook for another 1-2 minutes or until meat is white and pearly. Total cooking time is dependant on the size of the lobster tail. Internal temperature should read 140°F using a digital thermometer.
Remove and spoon the lemon garlic butter sauce over each tail, letting it seep down between the meat and the shell. Dust with smoked paprika.
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