Step 1

In a food processor, pulse one-third of shrimp into a smooth paste. Add remaining shrimp, scallions, cherry peppers, basil, 1 teaspoon salt, and 1/4 teaspoon pepper; pulse a few times just until coarsely chopped and evenly combined.

Step 2

Divide mixture into 4 portions; shape into patties slightly wider than buns and about 3/4 inch thick. Refrigerate on a parchment-lined plate until ready to grill or, loosely covered, up to 1 day.

Step 3

Preheat grill to medium-high. In a small bowl, whisk together mayonnaise, paprika, onion powder, cayenne, and lime juice; season with salt and pepper. Brush grates and shrimp patties with oil.

Step 4

Grill patties, flipping once, until just cooked through and charred in places, 3 to 4 minutes a side. Grill split sides of buns just until lightly toasted, about 1 minute.

Step 5

Dress split sides of buns with spicy lime mayo; top with lettuce, tomatoes, patties, and more cherry peppers and basil. Serve immediately.