Preheat the oven to 350 F (177 C). Line an 8 x 8 pan with parchment paper along the bottom and two sides.
Make the shortbread crust. Add the sugar and lemon zest to a bowl of a stand mixer fitted with paddle attachment or to a mixing bowl (to mix with a hand mixer). Whisk the sugar and lemon zest together until a sand like texture forms. Cream the butter into the lemon sugar at medium speed for about 4 - 5 minutes. Mix in the salt, vanilla extract and flour at low speed until crumbly and well combined.
1 cup (200 g) granulated sugar, 2 large lemons, zested, 1 cup (226 g) unsalted butter, room temperature, 2 cups (240 g) all purpose flour, ½ teaspoon salt , 1 teaspoon vanilla extract
Evenly press the shortbread mixture into the prepared pan. Press some of the dough about ½ inch up the sides of the pan to from a little crust. Bake the shortbread for about 20 - 25 minutes or until golden brown. Cool on a cooling rack while preparing the lemon raspberry filling. In another large mixing bowl, whisk the sugar and zest together until a sand like texture forms. Whisk in the eggs, followed by the salt and flour until smooth and well combined. Whisk in the freshly squeezed lemon juice.
1¼ cups (250 g) granulated sugar, 2 large lemons, zested, 3 large eggs, room temperature, ¼ teaspoon salt , ½ cup (60 g) all purpose flour, ½ cup (120 ml) freshly squeezed lemon juice
Top the mixing bowl with a fine mesh strainer. Press the fresh or thawed raspberries into the bottom of the strainer, allowing the raspberry juice to fall directly into the bowl. Whisk well to fully combine the raspberry juice with the lemon custard.
1½ cups (280 g) fresh or frozen raspberries
Pour the raspberry lemon filling into the prepared pan on top of the parbaked crust. Bake the assembled raspberry lemon bars for 18 - 22 minutes. The edges will be set and the center will have a slight jiggle (like a cheesecake) when the bars are done baking. Cool the bars in the pan at room temperature for 1 hour. Transfer the bars to the fridge to chill for an additional 2 - 3 hours or up to overnight. Slice the bars and dust with powdered sugar right before serving. Bars are best served from the fridge.