Line a 9x13'' pan or jelly roll pan with parchment paper, leaving an overhang on the sides for easy removal. If you don't have parchment paper, you can generously butter the pan.
Add 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 milk, one can at a time, while stirring constantly. You want to make sure the mixture maintains a steady heat and constant boil.
Stir the mixture constantly, scraping the sides occasionally until it reaches a firm ball stage (about 220-245 degrees F on a candy thermometer). It takes patience and time. If you don't have a thermometer, you can 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 220-245 degrees F / or the firm ball stage, remove from heat. Stir in vanilla.
Pour the hot caramel into the prepared pan, spreading it evenly. Let it cool at room temperature for about 2 hours, or until fully set. Once set, use the parchment paper overhang to lift the caramel from the pan.
Slice into small squares or rectangles with a sharp knife. Wrap each piece in wax paper or parchment paper to prevent sticking.