Preheat oven to 350°F. Grease a 9-inch square pan.
Using an electric mixer on medium speed, beat butter, sugar and vanilla together in a medium bowl until light and fluffy, about 1 minute. Scrape the bowl with a rubber spatula. Add the egg and beat until smooth, 10 seconds. Scrape the bowl.
Add the flour mixture and beat on low speed until blended, 10 seconds, stopping the mixer once to scrape the bowl.
Using floured fingertips or a spatula, spread the dough evenly over the bottom of the prepared pan. Set aside.
To prepare the filling: With the mixer on medium speed, beat butter, salt and vanilla together until light and fluffy, about 1 minute. Scrape the bowl with a rubber spatula.
Add sweetened condensed milk and beat on low speed until blended, 15 seconds. Scrape the bowl, then add corn syrup and mix until blended, 10 seconds. Add the egg and beat for 1 1/2 minutes on medium-high speed until the mixture is light and fluffy again. Scrape the bowl.
With the mixer on medium-low speed, add the flour and beat until mixed, 30 seconds.
Pour the filling evenly over the base. Bake on the center rack of the oven until the edges appear set but the center is still jiggly, 25 to 27 minutes. (Bars baked in an 8-inch pan may take longer.) Let cool to room temperature; then refrigerate until set, about 4 hours. Cut into 2 1/4-by-1 1/2-inch bars with a sharp thin knife, dipping it in hot water and wiping it clean before each new cut.
Place the bars on a plate. If there are any remaining after a few hours, refrigerate, covered with plastic wrap, with a sheet of wax or parchment paper between layers. If keeping longer than several days, freeze; bring bars to room temperature before serving.