Butter two 8" x 8" or 10 ½" x 6 ½" 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.
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.
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.
Cut the caramels into squares, and wrap in waxed paper.
