Preheat the oven to 350°F (175°C). Line a 9×13-inch baking pan with parchment paper, leaving some overhang on the long sides to lift the bars out later. Lightly spray the parchment and sides with nonstick cooking spray or brush with a little melted butter.

In a medium bowl, whisk together 1 1/2 cups all-purpose flour, 1/2 cup powdered sugar, and 1/4 teaspoon fine sea salt.

Add the cold, cubed unsalted butter and vanilla extract (if using) to the dry ingredients. Using a pastry cutter, fork, or your fingertips, cut the butter into the flour mixture until it resembles coarse crumbs with small pea-sized bits of butter that start to clump when squeezed.

Transfer the crust mixture into the prepared pan. Press it firmly and evenly into the bottom of the pan, using the bottom of a measuring cup to smooth and pack it down.

Bake the crust at 350°F (175°C) for 15–18 minutes, or until the edges are just turning lightly golden and the crust is set but not fully browned.

While the crust bakes, place the chopped strawberries in a blender or food processor and blend until very smooth. For a silky filling, you may push the puree through a fine mesh sieve to remove seeds, if desired. Measure out 3/4 cup strawberry puree for the filling; reserve any extra for another use.

In a large bowl, whisk together the granulated sugar and eggs until well combined and slightly lightened in color. Whisk in the freshly squeezed lemon juice and finely grated lemon zest.

Whisk the measured 3/4 cup strawberry puree into the lemon mixture until fully blended.

In a small bowl, whisk together 1/4 cup all-purpose flour, 1 tablespoon cornstarch, and 1/4 teaspoon fine sea salt. Sprinkle this dry mixture over the strawberry-lemon mixture and whisk gently just until smooth and no dry streaks remain. If desired, add a drop or two of red or pink gel food coloring and stir to reach your preferred shade.

When the crust is done, remove it from the oven but leave the oven on. Give the strawberry lemon mixture a final gentle stir, then carefully pour it over the hot crust, spreading evenly to the corners.

Return the pan to the oven and bake at 350°F (175°C) for 22–28 minutes. The edges should be set and slightly puffed, and the center should no longer look liquid but still have a slight jiggle, like firm gelatin, when you gently shake the pan.

Remove the pan from the oven and place it on a cooling rack. Let the bars cool completely to room temperature in the pan. Once cooled, cover the pan and refrigerate for at least 2 hours, or until fully cold and set.

Use the parchment overhang to lift the chilled slab of bars out of the pan and onto a cutting board. With a sharp knife, cut into 20–24 squares or rectangles, wiping the knife with a warm, damp cloth between cuts for clean edges. Just before serving, dust the tops with powdered sugar and garnish with sliced strawberries or thin lemon slices, if desired.