Whisk together mayonnaise, relish, and 1/4 teaspoon of the Old Bay in a small bowl.
Place half of the shrimp in a food processor and pulse until very finely chopped. Add egg, garlic, and remaining 3/4 teaspoon Old Bay; pulse just until combined. Transfer mixture to a medium bowl. Roughly chop remaining shrimp and add to bowl. Add parsley and stir just until combined. Shape mixture into four 3 1/2-inch patties.
Melt butter in a large skillet over medium-high heat. Add patties in a single layer; cook until edges are crispy and shrimp is cooked through, 4 to 5 minutes per side.
Place 1 patty on bottom half of each muffin; top with 4 teaspoons of the mayonnaise mixture, tomato slices, 1 lettuce leaf, and other half of muffin.