Combine mayonnaise, minced jalapeno pepper, cilantro, lime juice, sugar, lime zest, cumin, and 1/2 teaspoon salt in a bowl; stir until smooth. Chill until serving.
Combine water, chicken broth, sliced jalapeno peppers, 3 tablespoons salt, and lime in a large Dutch oven; bring to a boil. Add lobsters; cook until bright red, 8 to 10 minutes. Drain and cool until easily handled, 5 to 10 minutes.
Remove meat from lobster shells, tails, and claws. Serve with mayonnaise-cilantro mixture.