Lightly oil an 8 or 9 inch square baking pan or line with parchment paper. Set aside.
In a small sauce pot, heat the heavy cream until lukewarm. Set aside.
In a medium sauce pot, combine brown sugar, corn syrup and salt. Bring to a boil over medium high heat, stirring constantly to dissolve the sugar. Once it reaches a boil, stop stirring, reduce the heat and cook to firm ball stage or 248℉/120℃ on a candy thermometer. This will take about 1-2 minutes. If you don't have a candy thermometer, drop some of the mixture into a cup of very cold water. If it forms a firm ball, it is ready. (Do not worry if it forms a hard ball.)
Slowly begin adding the lukewarm heavy cream, stirring constantly. Do not allow the mixture to stop boiling. Once all the cream has been added, stop stirring and continue cooking on medium low heat until the mixture reaches 248℉/120℃ (firm ball stage) again. This will take much longer, around 35-45 minutes. Start checking the mixture after about 30 minutes with a candy thermometer or using the cold water test.
Once the mixture reaches 248℉/120℃ again, start adding the pieces of unsalted butter bit by bit so the mixture never stops boiling. Stir constantly; it will take a while for the butter to fully incorporate. Once all the butter has been added and is well mixed in, stop stirring and cook one last time to 248℉/120℃ or firm ball stage. This will take approximately 8 more minutes. Remove from heat.
Stir in vanilla until well mixed. Pour the caramel into the prepared baking pan, getting in as much of it as you can without scraping too much. Allow the caramel to cool and set up for a few hours. Once cooled, turn out onto a board and cut the caramel slab into 64 squares (or as many as you'd like). Wrap each square in waxed paper or other candy wrappers. Caramels will keep in a cool dry place for several months.