To make the strawberry filling, combine strawberries, sugar, lemon zest and lemon juice in a small saucepan. Set over moderate heat and bring to a simmer. Cook, stirring often, until the berries have softened and the sauce has thickened, about 8 to 10 minutes. Set aside and allow to cool completely.
To make the base, preheat oven to 350°F. Line a 10-inch square baking pan with parchment paper and set aside.
In a large bowl, mix together flour and powdered sugar. Add butter and mix until combined. Transfer to prepared baking pan and press down into an even layer. Bake until lightly golden brown, about 20 to 25 minutes. Remove from the oven and allow to cool.
To make the lemon filling, beat together the eggs, sugar lemon zest and lemon juice until thick and pale in color. Stir in the flour, baking powder and salt until just combined.
Pour the strawberry filling onto the cooled crust and spread out into an even layer. Then pour in the lemon filling. Give it a few light taps on the counter to remove any air bubbles. Bake until just set in the center. It should wobble slightly but not too much, about 25 to 30 minutes. Remove from the oven and allow to cool completely. Chill until ready to serve.
To make the topping, combine the freeze-dried strawberries and powdered sugar in a food processor. Pulse until the strawberries are well-blended into the sugar. Add to a sifter and dust the top of the bars completely with the strawberry sugar. Slice into 12 bars and serve. Enjoy!