Begin by cutting 3/4 of the way through the ridged back of the shrimp, removing the vein that runs down the center. Rinse and pat the shrimp dry.
The marinade: Whisk together the thyme, lime juice, 2 tablespoons of oil, salt, and pepper. Pour marinade into a shallow, nonreactive bowl and lay the shrimp cut side down in the mixture. Refrigerate for 30 minutes.
Preheat your grill or skillet to medium high.
In a food processor, combine the remaining tablespoon of oil, 1/2 teaspoon of salt, garlic, jalapeƱo, scallions, and pistachios. Pulse until it turns into a coarse paste. Add the herbs and pulse until it is finely chopped. Taste for seasoning and adjust as needed.
Spoon and pack the mixture into the shrimp; close the shrimp once filled. If using a skillet, add additional 2 tablespoons olive oil and cook the shrimp shell side down for 3 minutes on one side. Turn them over and cook on the other side for 2 minutes (if using a grill, cover it for the last 2 minutes). If your shrimp are smaller, they will require less time. Cook until just opaque, keeping in mind that the shrimp will continue to cook off the heat. Once ready, drizzle white balsamic on top.