Combine white sugar, 1/2 cup cream, light 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 (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 heat; stir in vanilla extract.
Pour caramel mixture into baking pan. Allow to cool slightly. Cover with plastic wrap and refrigerate, 8 hours to overnight.
Remove baking pan from the fridge 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 melting, about 1 1/2 minutes.
Cut caramel into 1-inch squares. Dip caramel squares in melted chocolate, shaking off excess. Place on lined baking sheet; sprinkle with sea salt. Chill until hardened, about 1 hour.