In a small (but not too tiny - you need some room to stir somewhat vigorously) bowl, use a fork to mash together the butter, tarragon, and Tabasco. Add a pinch of kosher salt and black pepper. Keep stirring until the butter mixture is uniform. Cover and set aside until ready to serve.
Pat the shrimp dry with paper towels, then season to taste with kosher salt and black pepper. Preheat your grill or grill pan to high heat. Grill the shrimp until just cooked through, about 3-4 minutes a side, depending on their exact size. You know they are done when the backs are opaque and they feel very firm when pressed with tongs. Remove to a wide shallow bowl.
Immediately use a spoon to chunk up the butter and add those chunks to the shrimp. Keep stirring and tossing until the butter is melted and the shrimp is generously coated.
Serve immediately with a torn baguette for dunking in the butter.