Heat oven to 400 degrees. Line a rimmed sheet pan with parchment paper.
Blot the fish dry with a paper towel. Mix the tarragon, oregano, pepper and lemon juice into the olive oil. Brush both sides and the cavities of the branzino with the olive oil mix.
Spread the tapenade into the cavities.
Place the lemon slices, tomato halves, capers and fresh herbs inside the fish. Be sure not to overstuff the fish, you will want the cavity to remain closed.
In a large bowl, stir the salt with ½ cup cold water, adding more water as needed. You want the salt to feel like damp (not wet) sand. You should almost be able to mold it.
Create a salt bed for your fish on the parchment lined sheet pan. Scoop about half the salt onto the baking sheet and pat it into an even ½-inch-thick bed (or 2 separate beds) on which to lay the fish. Place the fish on top and use your hands to pat the remaining salt all around the fish so everything is enclosed and sealed. You may need to make more of the salt mixture. The goal is to have a thick crust that completely encloses the fish so it can steam inside. You can leave the tail sticking out as you won't be eating it.
Find the thickest part of the fish, usually the nape and use an instant read thermometer to poke a hole through the salt crust to touch the top of the fish. This is so you can take its temperature without cracking the crust. You are not keeping the thermometer in, just creating a hole.
Bake for 25-35 minutes. The fish is done when its internal temperature reaches 135 degrees (use the hole you made to insert the thermometer to check). If you do not have a thermometer use the 35 minutes to bake. It's hard to overbake the fish due to the encasement of salt. Take fish out of the oven and let rest for 5 to 10 minutes.
Using a butter knife or metal spoon, crack open the crust, removing it entirely from the top of the fish and discard salt. Gently place fish on another clean pan (minus the salt). Then peel off and discard the skin on top (it’s too salty to eat). Using a spoon or spatula, carefully remove the fish flesh from the bones (and poke out fish cheeks if you like). Lift the fish skeleton, remove lemons and herbs, you can keep the tapenade, capers and tomatoes. Scoop out bottom fillets, taking care not to puncture the skin which you will remove before serving.
Drizzle fish fillets with olive oil, if you like, and serve with lemon wedges.