I know that CT is a shall issue state, but I also believe that one must first apply at the local Sherrif's office before getting a state license. It is a burdensome process, but amazingl allowed considering what a liberal state CT is.
Actually, it's the local police departments that first must issue a Local Carry Permit, then you apply to the State Police for a State Carry Permit. Many towns, and just about all the cities, only issue a few permits a year. I tried to get one in Danbury about 10 years ago, and got told that I was too late. It seems that the 25 permits that Danbury issued for that year were all gone. Danbury had a population of 55,000 at that time, this effectively prevented me from getting a hand gun permit in this state, until I moved anyway. When I ask if there was a waiting list, I was told NO.
Connecticut has a 2 week waiting period on long gun purchases, unless you have a pistol permit or a hunting license. Connecticut is not a gun owner friendly place, we also have a "Assault Rifle" ban that even extends to the guards at the states two nuclear power plants.