Place the ginger and garlic in a food processor and process until finely chopped. Add the coconut milk, 2 teaspoons of salt, and 1/2 teaspoon of pepper and pulse to combine. Add the shrimp and cilantro and pulse until well chopped, and the mixture is just shy of a puree. Make sure to leave some texture; do not puree.
To shape the burgers, use wet hands since the patty mixture is sticky. Handle lightly to keep the texture light and juicy. Shape the shrimp into 4 evenly sized burgers about 1-inch thick. Cover and chill for at least 30 minutes to let the flavors develop or as long as several hours.
When ready to cook, build a medium-high heat fire in a barbecue. Brush the burgers on both sides with olive oil and season generously with salt and pepper. Cook until golden brown and cooked through about 8-10 minutes total. Regulate the heat as needed so the burgers can cook without burning. To grill, oil the grate, and cook the burgers, covered, as above. Be careful not to overcook, or the burgers will be dry.
While the burgers cook, toast the burger buns on the outer perimeter of the grill rack. Brush the toast with mayonnaise (optional).
Stir all ingredients together for the Avocado-Mango Salsa and set aside.
To build the burgers, arrange tomatoes and salsa mix on 4 pieces of toast, and top with the burgers. Cover with the remaining toast and serve immediately with the remaining salsa on the side.