In the bowl of a stand mixer, combine the sugar and butter until smooth, then add flour and salt; mix until dough forms. (If you prefer to make the dough by hand, stir together flour, powdered sugar, and salt in a medium bowl; with a pastry blender, cut in the butter until mixture resembles coarse crumbs.)
Sprinkle the crumb mixture into the prepared baking pan. Using your hands, press to cover bottom of pan firmly (I used plastic wrap and a small rolling pin, like I do when making shortbread).
Bake for 15 minutes or until light brown (it will be partially baked), then remove the pan from the oven.
Make the filling
While the crust is baking, mix together the eggs and add in sugar, butter, salt, and vanilla; whisk together until well combined and smooth.
Pour filling on top of partially baked crust and bake until set, about 25-30 minutes.
Cool the bars completely in the pan on a wire rack.
Lift the uncut bars out of pan by aluminum foil or parchment paper. Transfer to a cutting board; cut into bars.
Dark brown sugar really emphasizes the molasses flavor, but you can use light brown sugar as wellWatch that the shortbread crust doesn’t get too dark when you're first baking it…at 20 minutes, my crust was medium-dark brownBake the filling for no more that 35 minutes…it will puff up but will settle down when coolingCut the bars using a pizza wheel…it’s much easier. If you use a knife, just wipe the blade after each time you slice