This burger is flush with citrus and island spices. It requires a bit of work to assemble and a gentle hand when turning, but the result is something memorable.
Place a large skillet over medium heat and add 1 tablespoon of the oil and 1 tablespoon of the butter. When the butter is melted, add the onion and bell pepper and cook, stirring often, until they soften, about 6 minutes. Raise the heat to high and add the chopped shrimp, cilantro, scallions, garlic, ginger, thyme, cayenne, allspice, and lime zest. Cook, stirring often, until the shrimp is almost cooked through, about 2 minutes. Add the lime juice, orange juice, and salt and cook 1 minute longer.
Transfer the shrimp mixture with a slotted spoon to a medium bowl. Reduce any liquid left in the pan until it thickens, then transfer to the bowl with the shrimp. Let the mixture cool.
Add 1/2 cup of the bread crumbs and 1 of the eggs to the shrimp mixture and mix together until well combined. Shape the mixture into 4 burgers about 3/4 inch thick and 4 inches across. The mixture will be loose. Arrange the burgers on a platter so they aren’t touching, cover gently with plastic, and refrigerate for at least 1 hour or up to 6 hours.
Put the flour in a large, shallow bowl or pie plate. Beat the remaining 2 eggs in a bowl. Put the remaining 1/2 cup bread crumbs in a large, shallow bowl or pie plate. Dredge the chilled shrimp burgers in the flour so they are lightly dusted on both sides. Dip them into the eggs, then dredge them in the bread crumbs so they are lightly coated.
Place a skillet, preferably cast-iron, over medium-high heat and add the remaining 1 tablespoon each oil and butter. Just when the butter stops sizzling, arrange the shrimp burgers in the pan so they aren’t touching and cook, uncovered, for 3 minutes, until a nice crust forms on the bottom. Turn carefully and cook about 3 minutes more.
Serve on the hamburger buns, topped with some mayo and your choice of hot sauce.
Note: These burgers are too delicate to cook on an outdoor grill.