Place the garlic in a mortar and pestle with a generous amount. Grind to a paste. Add 1 tbsp olive oil and blend together.
Preheat the broiler. Cover a rimmed baking sheet with foil and place the tomatillos on top, stem side down. Place under the broiler on the highest rack and broil 2-5 minutes, until charred on one side. Turn over and broil the other side for 2-5 minutes until charred on the other side.
Remove tomatillos from the heat and transfer to a blender, including any juice that may have accumulated. Add chilies, basil, mint, and tarragon and blend until smooth. Season with salt to taste.
Season the shrimp with salt and pepper. Heat the remaining 1 tbsp olive oil over medium-high heat in a heavy saucepan and add the shrimp. Cook on one side for 1 minute then use tongs to turn the shrimp over and cook on the other side for one minute. Remove shrimp from the pan and set aside.
Add garlic paste to the hot pan and cook, stirring, for 30 seconds to 1 minute, until fragrant. Add the tomatillo mixture, stir together, and lower the heat to medium-low. Simmer for 5 minutes, stirring constantly, until the sauce thickens slightly.
Return the shrimp to the pan, bring to a simmer, and cook for about 3 minutes, stirring often, until the sauce coats the shrimp and the shrimp is cooked through. Taste, season with salt and pepper as necessary, and serve.