Preheat the oven to 350°F. Line a 9x13 baking dish with parchment paper, leaving enough overhang to lift the bars out later.
In a mixing bowl, beat the softened butter and powdered sugar until smooth and creamy.
Add the vanilla extract and mix until combined.
Add the flour and salt. Mix until a soft dough forms.
Transfer the dough to the prepared baking dish and press it evenly into the corners. Smooth the surface with clean hands or the bottom of a measuring cup.
Lightly dock the crust with a fork.
Bake for 18 to 22 minutes, or until the edges are lightly golden and the center is set.
Remove from the oven and allow the crust to cool slightly.
While the crust bakes, prepare the lemon filling. In a bowl, whisk the eggs until smooth.
Add the granulated sugar and whisk until fully incorporated.
Add the lemon juice and lemon zest, then whisk again.
Add the flour and salt and continue whisking until completely smooth.
Pour the filling over the warm crust.
Gently tap the baking dish on the counter to release any air bubbles.
Return the dish to the oven and bake for 22 to 26 minutes, or until the center is set with only a slight jiggle.
Remove from the oven and cool completely at room temperature.
Refrigerate for at least 2 hours, or until fully chilled.
Lift the bars from the pan using the parchment paper overhang.
Slice into squares using a warm knife, wiping the blade clean between cuts.
Dust with powdered sugar and top with fresh strawberries, blueberries, and blackberries just before serving.