Make the crab cakes: Tear the bread into very small pieces or rub between your hands into small crumbs. Heat 1 tablespoon butter in a small nonstick skillet over medium heat. Add the breadcrumbs and a pinch each of salt and black pepper. Cook, stirring occasionally, until the breadcrumbs are toasted and crisp, about 3 minutes; transfer to a large bowl and let cool. Melt the remaining 2 tablespoons butter in the skillet; let cool slightly.
Add the egg, mayonnaise, chives, parsley, lemon zest, paprika, cayenne and melted butter to the breadcrumb mixture; stir to combine. Add the crab and gently stir until just combined. Line a baking sheet with parchment paper. Form the crab mixture into eight 21/2- to 3-inch patties; transfer to the prepared baking sheet and refrigerate 1 hour.
Meanwhile, grind the oyster crackers in a food processor to make fine crumbs; set aside.
Make the sauce: Whisk the mayonnaise with the lemon zest and juice, cayenne, 1/4 teaspoon salt and a few grinds of black pepper in a small bowl until smooth. Stir in the chives. Refrigerate until ready to serve.
Cook the crab cakes: Heat 1/2 inch vegetable oil in a large nonstick skillet over medium-high heat until shimmering. Press the crab cakes into the ground oyster crackers, turning to coat. Working in batches, add the crab cakes to the skillet and fry, turning, until golden, 4 to 5 minutes. (Adjust the heat if the crab cakes are browning too quickly.) Drain on a paper towel-lined plate. Serve the crab cakes on the buns with lettuce and sauce.