Line an 8-inch x 8-inch baking dish with parchment paper, leaving enough paper for it to come up the sides of the dish. Spray paper with a good coating of non-stick cooking spray. Set aside.
In a medium saucepan, combine brown sugar, heavy cream, butter, and 1/2 cup rum. Heat over medium-high heat until butter is melted and sugar is dissolved.
Bring mixture to a boil and continue to heat, stirring occasionally, until candy thermometer reaches 245 degrees F.
Remove candy from heat, stir in 1 tablespoon rum and vanilla. (The mixture will bubble when the liquids are added, be careful.)
Pour caramel mixture into the prepared dish. Sprinkle the top with sea salt over the top of the caramels - you just want a little salt all over the place, don't go crazy!
Let caramels sit out and cool for 90 minutes. Lift the parchment paper from the baking dish and transfer to a cutting board. Cut the caramels into 1-inch squares, or into small rectangles, and wrap in waxed paper. Twist the edges to seal to candy.
Notes
You can use light brown sugar in this recipe instead of the dark brown. The color of the caramels will just be a bit lighter.