To obtain a State Permit to Carry Pistols and Revolvers, you must first obtain a Local Permit.
Connecticut law requires you to respond to where you reside to apply for the State Permit. For non-state residents, contact the Connecticut State Police at www.ct.gov/despp and follow the Special Licensing and Firearms Unit link for instructions.
Download a Greenwich Temporary State Pistol Permit via this link. The application has all the instructions necessary to obtain the permit.