Bring sugar, corn syrup, water and salt up to a boil over high heat in a heavy-bottomed pot. Boil, brushing down the sides of the pot occasionally with a brush dipped in cool water, until sugar reaches a temperature of 275 ?F (hardball) registered on a candy thermometer. Remove from heat and stir in butter and peppermint.
Let sugar cool for 40 minutes without stirring.
Butter a marble or other smooth work surface. Scrape cooling candy onto work area and using a spatula or bench scraper, spread and scoop up candy until cool enough to handle. Butter your hands and pick up candy, stretching into lengths (let gravity or a friend help you). Keep stretching taffy and folding taffy over itself, until it becomes pale (no longer translucent) and shiny, about 5 to 10 minutes. The longer you stretch the taffy, the better!
Set taffy down and roll to about 24 inches (and 1 1/2 inches wide). Let cool for 30 minutes, then cut with buttered scissors into small, bite-sized pieces. Wrap candy individually in parchment squares and store in an airtight container.