There is no restricted/unrestricted. There is a "Permit to Carry Pistols and Revolvers."
There is no requirement to show need. The application asks for a purpose, for which I put "all legal purposes."
Most everything you need to know is here:
http://www.ct.gov/dps/cwp/view.asp?A=2158&Q=294502
Regarding suitability (from that link):
Applicants must be at least 21 years old and be a legal resident of the United States. Persons convicted of a felony or any one of 11 misdemeanor offenses detailed in the statute are ineligible to receive a permit. Persons convicted as a delinquent for the commission of a serious juvenile offense (as defined in section 46b-120); persons discharged from custody within the preceding twenty years after having been found not guilty of a crime by reason of mental disease or defect (pursuant to section 53a-13); persons confined in a hospital for persons with psychiatric disabilities (as defined in section 17a-495), within the preceding 12 months by order of a probate court; persons subject to a restraining or protective order issued by a court in a case involving the use, attempted use or threatened use of physical force against another person; persons subject to a firearms seizure order issued pursuant to C.G.S. 29-38c after notice and hearing; and any person who is an alien illegally or unlawfully in the United States, are prohibited from obtaining a pistol permit.
The 11 misdemeanor offenses mentioned are those in subsection (c) of 21a-279, section 53a-58, 53a-61, 53a-61a, 53a-62, 53a-63, 53a-96, 53a-175, 53a-176, 53a-178 or 53a-181d, all of which can be looked up here:
http://www.cga.ct.gov/2009/pub/titles.htm
The short titles of them are:
Sec. 21a-279. Penalty for illegal possession
Sec. 53a-58a. Negligent homicide with a motor vehicle
Sec. 53a-61. Assault in the third degree
Sec. 53a-61a. Assault of an elderly, blind, disabled, pregnant or mentally retarded person
Sec. 53a-62. Threatening in the second degree
Sec. 53a-63. Reckless endangerment in the first degree
Sec. 53a-96. Unlawful restraint in the second degree
Sec. 53a-175. Riot in the first degree
Sec. 53a-176. Riot in the second degree
Sec. 53a-178. Inciting to riot
Sec. 53a-181d. Stalking in the second degree