Prepare a charcoal grill or preheat your gas grill to medium-high heat (375°F) and scrape clean.
Prepare the lobster tails by using kitchen shears to cut through the top of the shell lengthwise, splitting the tail in half down the middle. Now press down the split lobster tail flat on a cutting board and cut the tail completely in half with a larger “chefs” knife.
Gently detach the meat from the shell to expose the meat, and then lay back down in the shell. This is to ensure the lobster tail will not stick to the shell once grilled.
Pat lobster tails dry, brush with avocado oil, and season with salt and pepper.
In a small pot, mix the butter, minced garlic, lemon juice and zest, chopped parsley, chopped tarragon, salt, and pepper. This can be done on the grill, as well.
Place the lobster tails on the preheated grill, meat side down. Grill for about 2-3 minutes.
Flip the lobster tails over, shell side down, and brush with herb butter generously over the meat, making sure to get some into the shell as well.
Close the grill, and continue grilling for an additional 3-4 minutes, or until the lobster meat is opaque and firm.
Remove the lobster tails from the grill and transfer them to a serving platter. Allow them to rest for a couple of minutes.
Brush the grilled lobster tails with more herb butter.
Serve the grilled lobster tails with lemon wedges on the side to squeeze over the meat before eating. You can also serve them with extra melted butter for dipping, if desired.