You can speak to the building department in Brookhaven and they will be able to pull the file for that location and tell you if there are any open permits. That can be reviewed while you are there. This is an easy task to complete!
Information about permits can be found by going to the town building department, located at 1 Independence Hill in Farmingville, NY. As far as how long it takes to be resolved, it depends on the circumstances. If the structure is no longer there, why are we worried about a permit? A permit for nothing? If you need assistance please call me at 631-398-9543 and I would be happy to help.