Combine white sugar, 1/2 cup cream, corn syrup, and butter in a saucepan. Bring to a boil over medium heat, stirring often. Stir in remaining 1/2 cup cream. Cook, without stirring, until mixture reaches 242 degrees F (117 degrees C), or until a small amount of the caramel mixture forms a firm yet pliable ball when dropped in cold water. Remove from the heat and stir in vanilla.
Pour caramel mixture into the prepared baking pan. Allow to cool slightly, then cover with plastic wrap and refrigerate, 8 hours to overnight.
Remove the baking pan from the refrigerator and allow caramel mixture to warm up slightly, about 30 minutes. Line a baking sheet with waxed paper.
Combine dark and semisweet chocolate chips with shortening in a microwave-safe glass or ceramic bowl. Microwave in 30-second intervals, stirring after each interval, until melted, about 1 1/2 minutes.
Cut caramel into 1-inch squares. Dip caramel squares in melted chocolate, letting any excess drip off. Place onto the prepared baking sheet and sprinkle with sea salt. Chill until hardened, about 1 hour.