Line a 9x13'' pan or jelly roll pan with parchment paper. (If you don't have parchment paper you can generously butter the pan.) Either size pan will work--9x13'' will yield thicker caramels.
Add 2 sticks butter, sugar, and karo syrup to a large heavy-bottom saucepan over medium heat. Stir over medium heat until mixture begins to boil, about 5-10 minutes.
Gradually add the evaporated milk, one can at a time, taking about 12-15 minutes PER CAN to slowly add it, while stirring constantly. You want to make sure the mixture maintains a constant boil, otherwise your caramels can curdle.
Stir the mixture constantly, scraping the sides occasionally until it reaches a firm ball stage (about 240-245 degrees F on a candy thermometer). It takes patience and time, but it's sooo worth it! (Also, I don't really trust or rely on a candy thermometer--I like to test it the old fashioned way. Drop a spoonful of hot caramel sauce into a cup of ice water and mold it with your fingers into a ball. When ready it will feel pretty firm and pliable, but still slightly sticky.)
Once you reach 240-245 degrees F / or the firm ball stage, remove from heat. Stir in vanilla.
Pour caramels into prepared pan. Refrigerate until cooled and hardened. (Best if you refrigerate them overnight, or for several hours. They will be easier to cut and wrap).
The caramel will seem hard in the fridge after they're set, but they should be soft at room temperature. Cut caramel into small pieces and, if desired, wrap like a tootsie roll in wax paper.