I saw this on PBS on 'America's Test Kitchen' a couple of days ago.
They tasted shrimp burgers all up and down the North and South Carolina
coast and settled on the shrimp burgers from the Shrimp Shack on Saint
Helena Island (where I live). Here's what they came up with for
replicating at home without having to deep fry them:

1 c. Panko breadcrumbs, pulverized in a food processor

You want the crunch but not the sharp texture of Panko. It's not a
filler, it's for dredging. Set the crumbs aside.

Combine in the food processor:

1 c. large shrimp (about 21 ct.) peeled and deveined, tails removed
2 Tbs. mayonnaise
1/8 tsp. cayenne pepper
1/2 tsp. salt

Pulse these ingredients together but don't let it turn into a paste.
You want to still have discernible pieces of shrimp. Turn the mixture
out into a mixing bowl. Add three finely chopped scallions and mix
well. Form into 3 inch round burgers, about 3/4 inch thick. Dredge
them in the Panko and let them set for a few minutes.

Pan fry the shrimp burgers in about 3 Tbs. of neutral oil over medium
heat, about 3-5 minutes on each side until golden. Using two spatulas
makes turning them easier.

Spread some tartar sauce on the bottom of a hamburger bun. Add a shrimp
burger. Top the burger with a little more tartar sauce and a leaf of
Bibb lettuce. Voila!

This is one of the few restaurants in this area that actually serve hush
puppies. They were much more common on menus when I lived in the Mid-South.

Jill