Rinse salmon and pat dry; cut into 1/2-inch chunks. Pulse several times in a food processor to coarsely grind, scraping down container sides once or twice between pulses. In a bowl with a fork, mix ground salmon with broth, bread crumbs, dill, salt, and egg until mixture is thoroughly blended.
Make four foil strips 5 to 6 inches wide by 10 inches long. Fold each strip to form 5- to 6-inch squares, then fold in edges to seal layers. Coat one side of each foil square with vegetable oil. Divide salmon mixture into four equal portions, place a portion on oiled side of each foil square, and shape each into an evenly thick 4-inch-wide round. If making ahead, lay patties (on foil) in a single layer on a plate, cover airtight, and chill up to 1 day.
Place salmon patties on foil on a barbecue grill over a solid bed of medium coals or medium heat on a gas grill (you can hold your hand at grill level only 4 to 5 seconds); cover barbecue and open vents. Cook patties until firm when pressed and pale pink in the center (cut to test), about 10 minutes. When patties are almost done, lay rolls, cut side down, on grill and toast until golden, 1 to 2 minutes.
Spread the toasted side of each bun half with smoked salmon cream cheese. Layer bottom of each bun with lettuce and a salmon patty; top with cucumber-hearts of palm relish and set bun tops in place.
Cucumber-hearts of palm relish: In a bowl, combine 1/2 cup diced (1/4 in.) seeded English cucumber, 1/2 cup thinly sliced canned hearts of palm, 1/4 cup rice vinegar, 2 tablespoons chopped green onions (including green tops), 2 teaspoons sugar, and 1/2 teaspoon salt (or to taste). Mix and let stand at least 5 minutes or cover and chill up to 1 day. Serve with a slotted spoon. Makes about 1 cup.