Put 12 oz shrimp in a food processor and pulse into a rough paste for 30 seconds.
Transfer the shrimp to a big bowl. Mix the shrimp with 1/8 tsp salt, 1/2 tsp black pepper, 1/2 tsp garlic powder, 1/2 tsp paprika, 2 tbsp coriander, 1/3 cup breadcrumbs, and 1 large egg.
Shape the mixture into 4 patties, about 4 oz for each one.
Make the sauce: In a small bowl, whisk together 1 tbsp lemon juice, 2 tbsp Greek yogurt, 1 1/2 tbsp mayonnaise, and 2 tbsp parsley. Set aside.
Split 4 buns open. Heat a griddle pan on low heat and add 1/2 tbsp olive oil and 1/2 tbsp butter to the pan. Place the cut side down of 2 buns on the pan and grill until lightly golden brown, about 1 minute. Flip to the other side and continue to toast for another 1 minute before transferring them to a plate. Heat 1/2 tbsp olive oil and 1/2 tbsp butter in the pan and grill the remaining 2 buns. Set the buns aside.
Heat 1 tbsp olive oil and 1 tbsp butter over medium-low heat. Cook the burger patties, about 5 minutes on each side or until golden brown.
To serve, spread the buns with the mayonnaise sauce. Then top with 4 oz sliced tomatoes, 2 oz carol lettuce, and shrimp patties. Sprinkle 0.5 oz cheddar cheese and cover with top bun halves. Enjoy!