Preheat a grill to medium high (about 375 degrees F). In a food processor fitted with a steel blade, add 2/3 of the shrimp.
Add the garlic, paprika, mustard, salt, seafood seasoning, and cayenne. Pulse a few times to begin to incorporate. Add 1/2 of the green onion and 1/2 of the bell pepper. Puree until fairly smooth, stopping to scrape down sides of bowl as needed. Transfer to a large mixing bowl.
Finely chop the remaining shrimp, bell pepper, and green onion (you want the shrimp to bind with the burgers but still be a little chunky so that the burgers have texture). Add to the bowl and stir until evenly combined.
Shape mixture into 6 patties that are about ¾-inch thick and 3 1/2 to 4 inches in diameter. Place in the refrigerator and chill for 15 minutes.
Make sure your grill is very clean. Brush the outsides of the shrimp patties with oil. Place on the grill (clean it one more time for good measure—any debris can make the patties stick), close the lid, then cook on the first side without disturbing for 4 to 5 minutes, until the patties release easily. Flip and cook on the other side until the patties are cooked through, about 3 to 4 minutes more.
If desired, in the last few minutes of cooking, place the buns on the grill, cut side down, and lightly toast.
Serve the shrimp burgers on the toasted buns, topped with Greek yogurt, mustard, lettuce, tomato, and avocado as desired. Enjoy!
Notes
TO STORE: Place cooked burgers in an airtight storage container or individually wrap and refrigerate for up to 2 days.
TO REHEAT: Very gently reheat burgers in a nonstick skillet on the stovetop over medium-low heat.
TO FREEZE: Individually wrap cooked or uncooked burgers, and freeze them in an airtight freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating or cooking.