Butter two 8" x 8" or 10 1/2" x 6 1/2" pans. Measure the vanilla into a small container, and set it aside for later., Cut the butter into 1" chunks, to help it melt faster. Combine the cut-up butter, half & half, corn syrup, sugars, and salt in a heavy-bottom, 5- to 6-quart saucepan., Heat the mixture over medium-low heat until everything has melted, stirring constantly. Attach a candy thermometer to the side of the pan., Cook slowly, stirring constantly, until the mixture reaches 248°F, or until a small amount dropped in cold water forms a firm ball. As the mixture boils it will become thicker and thicker and the frothy bubbles will give way to a more blurpy bubble that resembles thick lava. This happens gradually, but you'll notice it around 230°F and higher. This is when you really have to pay attention and watch the temperature carefully, as it tends to move more quickly from 230°F up. The whole process will take about 2 hours., Remove the pan from the heat, and carefully stir in the vanilla; the mixture will bubble and boil., Quickly pour the caramel into the prepared pans. Don't add any residue from the pot into the pans; instead, scrape it into a separate container. Baker's treat!, Wait 30 minutes, then sprinkle with coarse-flake sea salt., Cover the candy securely, and refrigerate., Remove the candy from the pan by loosening its edges with a knife or offset spatula, and banging the pan on a cutting board until the slab releases from the pan. Allow the candy to warm up a bit if it's hard, or it'll shatter when you bang the pan., Cut the caramels into squares, and wrap in waxed paper. Store in the refrigerator for longest shelf life. Wrapped caramels can also be frozen.