Preheat the oven to 350 degrees F. Combine the mayonnaise, minced pickles, shallot and hot sauce in a small bowl; season with salt and pepper. Set aside.
Toss the shrimp with the Cajun seasoning and 1/4 teaspoon salt in a large bowl. Melt the butter in a large nonstick skillet over medium-high heat. Add the shrimp and cook until pink, 1 1/2 to 2 minutes per side.
Meanwhile, arrange the rolls on a baking sheet and toast in the oven, about 5 minutes. Spread the mayonnaise mixture on the cut sides of each roll. Fill with the shrimp, lettuce and tomatoes. Serve with pickles.