In a bowl, combine all seasonings & add ground beef & beaten egg. Lightly combine without overmixing. Form into 4 burger patties. Set aside in refrigerator.
Shrimp
Peel & devein shrimp. Rinse in cold water & dry with paper towel. Toss shrimp in olive oil & Old Bay seasoning. Set aside in refrigerator.
Avocado Steak Sauce
In food processor, add all sauce ingredients. Process on high speed until smooth. Remove from processor & place in a glass dish. Cover with plastic wrap & refrigerate.
Other Ingredients
Slice tomato & red onion. Slice or shred cheese. Very lightly butter hamburger buns.
Cooking / Assembly
Heat a griddle to medium-high & cook shrimp until they are just done, about 1-2 minutes per side. Set them aside.
Place burgers on the griddle & cook them until done or to your liking.
Assemble surf & turf burgers as follows: roll, burger, cheese, arugula, tomato & onion, generous amount of avocado steak sauce, shrimp & top of roll.