Fill a stockpot with water and bring it to a boil.
As the water comes to a boil, heat a deep skillet or large sauce pan on medium low heat. Add the unsalted butter to the pan and let it melt. Make sure the butter is not bubbling and that the pan is not too hot, or else the butter will brown or burn. Low is best!
Adjust the heat to low, then add the shallots, garlic, parsley and tarragon to the butter and let it infuse for 15-20 minutes while you continue with the rest of the recipe.
Prepare an ice bath in a large bowl and place it on the counter as close to the stockpot of boiling water that you can.
Poach the lobster tails for 2 minutes then immediately put them into the ice bath. Let them come to temperature for a few minutes until they are comfortable to handle with your hands (probably a few minutes.)
Remove the lobster tails from the water. With kitchen shears, cut down the shell on the back side of the lobster tail. Apply pressure with your hands on the sides of the cut shell to help release the meat from the shell. The lobster tail should easily come out of the shell. Repeat with all of the lobster tails.
With a paring knife, make a shallow slice all the way down the center of the back of the lobster tail. You will probably see a gross looking vein (their digestive tract, yikes!) Remove that vein. Repeat with all of the lobster tails. Then, rinse all of them under cold water to get any excess gunk off of them.
Return to your butter. Taste to make sure the aromatics and herbs have adequaltely infused into the butter. If you want it to taste more like your aromatics, leave it for another 10 minutes or so. Then, add 1-2 generous sprinkles of salt to taste. I recommend doing one pinch, then mixing and tasting. If it's not salty enough, go with a little more. You do not want to over salt this butter as it's the base of the sauce as well. You will know when it tastes "just right" for you!
Put a fine mesh strainer over a large bowl. Pour all of the butter into the strainer, straining out the aromatics and milk solids. Pour the clarified infused butter back into your deep skillet. If you're making your own lobster broth for the sauce, feel free to pop those aromatics into the boiling water with the shells.
Put the skillet with the butter onto a burner. Adjust the heat to medium/medium low. Place the lobster tails into the butter, on their sides. Poach for 3-4 minutes. During this poaching time, use a large spoon or ladel to pour the hot butter onto the tops of the lobster tails. After 30 seconds, rotate the lobster tails, flipping so the exposed side now gets submerged into the butter. Continue to rotate them and baste them throughout the entire cook time. Remove the tails when they reach 135-138 degrees F internally. They will residually cook and reach 145 F temperature when they are removed.
Pour 1/2 cup of the infused butter into a sauce pan. Turn the heat on medium high. Add 2 tbsp of all purpose flour and whisk until lightly bubbly - about 1-2 minutes.
Combine the wine, seafood stock and lemon juice in a cup. Slowly pour the wine/stock mixture into the sauce pan while simultaenously whisking the roux. Let the sauce reduce and thicken for 2-3 minutes. Add the half and half and ocntinue to whisk. Taste the sauce. If it needs salt, season with light pinches of salt to taste until you reach the desired level. Once the sauce hits *different* remove it from the heat.
Right before plating, heat the butter in the skillet on the stove and pop the lobster in for a quick 1 minute warm-up.
Pour the sauce into a large shallow bowl, then plate the tail on top. Drizzle more sauce onto the tail and top with finely chopped fresh parsley.