Preheat broiler.
Put lump crab meat in a bowl but do not shred. Add 1 tablespoon lime juice and toss to combine.
In a small bowl whisk together remaining tablespoon lime juice, mayonnaise, and mustard until smooth. Pour sauce over crab and toss to coat. Season crab mixture with salt and pepper and chill, covered, 30 minutes.
Lightly toast brioche and with a 3 1/4-inch round cutter cut out rounds. Lightly butter toast. Spoon one fourth crab mixture into a 1/2-cup measure. Holding a toast round on top of mixture in measure, invert crab onto round and set on an ungreased baking sheet. Repeat procedure with remaining crab mixture and rounds and sprinkle 1 tablespoon Parmesan over each sandwich. Broil sandwiches about 3 inches from heat until cheese is melted and golden, 1 to 2 minutes.
Serve sandwiches with lime wedges and garnish with chives.
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