Increase the oven temperature to 350°F.
In a large bowl, or the bowl of a stand mixer fitted with the paddle attachment, combine the sugar, flour, lemon zest, salt and vanilla. Whisk in the eggs, lemon juice and milk until fully blended.
Pour the lemon mixture over the crust.
Bake until the lemon filling is set in the center and it no longer wiggles when the pan is moved, about 30-35 minutes. Remove before the lemon bars are lightly browned.
Remove and cool on a wire rack until room temperature and then chill for 1-2 hours.
Carefully remove the bars using the parchment sling. Place onto a cutting board and slice into 16 bars, cleaning the knife between each slice.
Right before serving, dust with powdered sugar.