Add the shrimp to a food processor. Pulse until finely chopped. They should not be pureed completely, there should still be some texture.
In a large bowl, mix together the shrimp, breadcrumbs, cheese, onion, garlic, egg, mayo, dill, worcestershire sauce, lemon juice, lemon zest, and spices until combined.
Divide the mixture into 6 portions. Shape each into burger patties.
Heat a skillet over medium-high heat. Add butter and sear patties on both sides until golden brown (3-4 minutes per side).
Transfer seared patties to a preheated oven (350°F/180°C) for 8-10 minutes to ensure thorough cooking.
Mix together all of the ingredients for the sauce.
Toast the burger buns. Place the shrimp patty on the bun and add preferred toppings and sauce, like lettuce, tomato, and avocado. Serve hot.