Grilled Lobsters with Southeast Asian Dipping Sauce

Bring an 8-quart kettle three fourths full with water to a boil for lobsters.
Mince garlic and in a bowl stir together with fish sauce, lime juice, brown sugar, water, and chili paste. Dipping sauce may be made up to this point 1 day ahead and chilled, covered.
Plunge 2 lobsters headfirst into boiling water and cook 3 minutes (lobsters will be only partially cooked). Transfer lobsters with tongs to a colander to drain and cool. Return water to a boil and cook remaining 4 lobsters in same manner.
When lobsters are cool enough to handle, twist off tails and break off claws at body of each lobster, discarding bodies. With kitchen shears halve tails (including shells) lengthwise. (Do not remove tail or claw meat from shells.) Lobsters may be prepared up to this point 1 day ahead and chilled, covered. (If making ahead, chill lobster immediately and keep chilled until ready to grill.)
Prepare grill.
Chop cilantro and mint and stir into dipping sauce. Measure out 1/4 cup dipping sauce to use for basting lobster tails. Grill claws on a rack set 5 to 6 inches over glowing coals, covered, turning them occasionally, until liquid bubbles at open ends, about 5 minutes, and transfer to a serving platter.
Arrange tails on grill, cut sides up, and brush with basting sauce. Grill tails, covered, basting occasionally, 6 minutes, or until meat is opaque.Serve lobster with dipping sauce and lime halves.