A trifunctional aziridine linker, trimethylolpropane tris(2-methyl-1-aziridinepropionate) (TTMAP), was melt blended with linear polylactic acid (PLA) to make star branched PLA. Adding pyromellitic dianhydride (PMDA) led to long chain branched (LCB) PLA. Mixing torque evolution during melt processing revealed high reactivity of aziridine with the carboxyl end group on PLA and an incomplete reaction of PMDA with the hydroxyl end group. Star-shaped PLA exhibited higher viscosity but no strain hardening in extensional flow while LCB PLA showed significant extensional hardening. Excess TTMAP in the branching reaction resulted in gel formation, which led to failure at low strain in extension. PMDA conversion was estimated based on gelation theory. The strain rate dependence of extensional hardening indicated that the LCB PLA had a low concentration of long chain branched molecules with an H-shaped topology. Unlike current methods used to branch PLA, free radical chemistry or use of an epoxy functional oligomers, our branching strategy produced strain hardening with less increase in shear viscosity. This study provides guidelines for design of polymers with low shear viscosity, which reduces pressure drop in extrusion, combined with strong extensional hardening, which enhances performance in processes that involve melt stretching.