Plunge a large, sharp knife into top of each lobster body just behind head and cut through head to kill lobster quickly. Cut off antennae and reserve. Holding lobster body in one hand, twist off tail with the other, then pull off claws where they join body. Cut tail in half lengthwise, pry meat out of shells, cut into H" cubes, and set aside. Cut through joint just below claws to separate claws from elbows; set claws aside. Save bodies, tail shells, and elbow joints for use in the lobster broth, if you like.
Put cilantro, scallions, shallots, lemongrass, and chile into a gallon-size zipper-top plastic bag, squeezing out as much air as possible, and place bag, unsealed, on a work surface. Pound bag with a mallet or flat side of a knife or cleaver until aromatics are well crushed. Cut open bag and, using a rubber spatula, scrape aromatics into a medium heavy-bottomed pot. Add tamarind pulp, tomato paste, and broth and bring to a boil over high heat. Reduce heat to medium-low and simmer, stirring occasionally, until bisque is reduced by one-quarter (to about 6 cups), 30–40 minutes.
Meanwhile, bring a medium pot of salted water to a boil over medium-high heat. Add reserved lobster claws and cook until bright red and claw meat is cooked through, 3–5 minutes. Drain, then cool under cold running water. Using the dull side of the knife, crack shells, then remove and discard shells from claws, leaving pincer shells in place, and set claws aside. Heat oil in a medium skillet over medium heat. Add reserved tail meat and cook, stirring frequently, until just opaque, about 30 seconds. Transfer with a slotted spoon to paper towels to drain and set aside.
Strain bisque though a fine sieve into a medium bowl, discarding aromatics. Return bisque to cleaned pot and bring to simmer over medium heat. Stir in sugar and cayenne and season to taste with salt.
Put reserved lobster claws into a small pot of boiling water over high heat and boil until heated through, about 30 seconds. Drain, then skewer each claw onto a reserved antenna. To serve the bisque, divide the reserved tail meat between 4 narrow soup bowls, ladle hot bisque into bowls and garnish by placing and antenna across each bowl with pincer shell resting on edge of bowl and meaty end of claw submerged in bisque.