Combine the butter and the Tabasco sauce in a small saucepan over medium-low heat, and cook until the butter has melted. Stir, and let cool slightly. Then divide the butter mixture between two large mixing bowls.
Place the shrimp in one bowl and coat them well with the butter mixture. Cover the bowl loosely and let the shrimp marinate at room temperature for 20 minutes.
Meanwhile, make the blue cheese dipping sauce. Combine all sauce ingredients, except the cheese, in a bowl. Gently fold in the blue cheese, being careful not to break it up too much (or it will turn the sauce an unappetizing gray). Adjust the seasonings to taste. Refrigerate, covered, for as long as overnight.
To cook the shrimp, heat a nonstick skillet over medium heat. Using tongs, lift the shrimp from the marinade and cook them, in batches, in the skillet until they are pink and opaque, 2 minutes per side. As they are cooked, transfer the shrimp to the second bowl and toss them well in that batch of butter-Tabasco mixture.
Again using tongs, transfer the cooked shrimp to a serving bowl. Sprinkle them with coarse salt. Serve immediately, along with the celery, carrots, and scallions, and a bowl of Blue Cheese Sauce for dipping.