Move the oven rack to the middle, and preheat oven to 350 degrees F. Line an 8-inch square baking pan with aluminum foil or parchment paper, leaving a nice overhang on all four sides.
Add the flour and sugar to the bowl of a food processor and pulse to combine.
Add the butter and pulse until the dough almost looks like a cohesive dough, this will take about 15-20 1-second pulses.
Press the dough into an even layer into your prepared pan. Bake the shortbread crust until it’s lightly golden in color, about 15 to 20 minutes. Move the crust to a wire rack to cool. Leave the oven on.
MAKE THE LEMON FILLING:
While the crust cools, in a medium saucepan whisk together the eggs, egg yolks, granulated sugar, and salt. Add the lemon zest and lemon juice and cook over medium-low heat, while stirring constantly with a rubber scraper. Keep stirring until the filling is thickened and pudding-like, 8-10 minutes.
Place a fine-mesh strainer over a medium-size bowl. Pour the lemon filling into the strainer and use your rubber scraper to press the mixture through. Make sure you scrape the underside of the strainer into the bowl.
Stir in the butter until it’s melted.
Pour the filling over the slightly cooled crust and spread into an even layer.
Bake the lemon bars until the filling is set, about 15-20 minutes. Place the pan on a wire rack and cool to room temperature, about 1-2 hours.
Use the foil or parchment overhang to lift the lemon bars from pan. Use a knife to cut the bars into 9 squares.
Dust the lemon squares with powdered sugar just before serving.