Preheat oven to 350 degrees Fahrenheit.
To make the crust: Combine the flour, powdered sugar, salt and butter in a food processor. Pulse until mixture resembles fine crumbs. Press into a greased 13x9-inch pan and bake for 18 to 20 minutes, or until golden brown. (Optional: Line pan with parchment paper sticking out of the sides to easily lift the bars out of the pan after they are baked.)
Meanwhile make the filling: In a food processor or blender, pulse the lemon zest (optional), lemon juice, and strawberries until smooth.
Whisk together the sugars, flour, and salt in a bowl. Add the eggs and yolk, one at a time, whisking well after each addition. Add the strawberry-lemon puree and mix until everything is smooth and combined.
Pour filling over hot crust and return to oven to bake for an additional 18 to 22 minutes. When done, the filling will no longer jiggle and a toothpick will come out clean when pressed into the center.
Let bars cool to room temperature in the pan. Chill in the refrigerator for 2-3 hours before serving. If parchment was used, lift bars out of the pan before cutting. Dust with powdered sugar if desired. Store leftovers in the refrigerator.
* Choose firm strawberries over soft, super ripe berries. The more ripe, the more moisture, which may produce a softer filling or cause trouble setting.