Step 1Heat the oven to 350 degrees.
Step 2In the bowl of a stand mixer fitted with the paddle attachment, or in a bowl using a hand mixer, beat the butter and sugar until creamy, 3 to 4 minutes. Mix the flour and baking powder together and beat into the butter mixture.
Step 3Press the dough into the bottom of a 13- by 9-inch baking dish. Bake until golden, 30 to 35 minutes. Cool to room temperature.
Step 1In a large saucepan, combine the sugar, water and corn syrup, stirring until the sugar has the consistency of wet sand. Place the saucepan over high heat and cook until the sugar dissolves and begins to boil. Do not stir the sugar, as this may cause it to seize.
Step 2While the sugar is cooking, combine the cream and butter in a saucepan over medium heat. Keep an eye on the sugar while you're heating the cream to keep it from scorching. Cook until the butter melts, stirring it into the cream. When the mixture has come to a simmer, remove from heat.
Step 3Continue to cook the sugar until it darkens to a rich caramel color, 7 to 10 minutes — the sugar will darken quickly and noticeably and will smell faintly nutty. Swirl the pan as the sugar darkens to judge the true color of the caramel (the sugar may darken in patches if there are hot spots on the stove). Watch carefully, as the sugar can easily overcook at this point and burn.
Step 4As soon as the color is darkened to a rich caramel, remove the pan from the heat and quickly add the cream mixture in a slow, steady stream. The sugar will bubble and steam as the cream is added; be careful as both the mixture and steam are very hot. Carefully stir in the vanilla, then continue to stir until the mixture stops bubbling. Remove from heat and set aside until cool to just warm.
Step 5Pour the cooled caramel over the shortbread, using a spatula to push it to the edges of the dish for an even layer. Sprinkle over the coarse salt. Refrigerate until cold, then cut into bars.