Process half of the shrimp in a food processor until a paste forms. Add cod, remaining shrimp, the breadcrumbs, capers, scallions, parsley, lemon juice, salt, and pepper. Pulse until just combined.
Using dampened hands, shape into 4 patties. Refrigerate, covered, until cold and firm, about 1 1/2 hours (or overnight).
Preheat grill to medium-high (if using a charcoal grill, coals are ready when you can hold your hand above grill for just 3 to 4 seconds). Brush grill and burgers with oil. Grill, flipping once, until cooked through, about 3 minutes per side. Toast buns on grill during the last minute of cooking, if desired. Serve burgers on buns; top with lettuce. Serve with tartar sauce, if desired, and the lemon wedges.
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