Preheat your oven to 350°F (175°C) and line an 8x8-inch baking pan with parchment paper, leaving some overhang for easy removal later.
In a mixing bowl, combine the flour, granulated sugar, and salt for the cookie base. Add the cold cubed butter and use a pastry cutter or your fingertips to cut the butter into the dry ingredients until the mixture resembles coarse crumbs.
Transfer the cookie base mixture into the prepared baking pan and press it down evenly using the back of a spoon or your fingertips to create a compact layer.
Bake the cookie base in the preheated oven for about 15 minutes or until it turns light golden brown. Remove from the oven and set aside to cool slightly.
In a blender or food processor add fresh raspberries and water. Blend until smooth then use a fine mesh strainer to strain the raspberry liquid without any seeds.
In a separate bowl, combine the raspberry puree, granulated sugar, flour, lemon juice, and lemon zest for the lemon raspberry layer. Gently stir the ingredients together until well mixed.
In a small bowl, lightly beat the eggs. Add the beaten eggs to the raspberry mixture and stir until all the ingredients are well combined.
Pour the raspberry lemon mixture over the baked cookie base, spreading it out evenly.
Place the baking pan back into the oven and bake for an additional 25-30 minutes or until the top layer is set and lightly golden.
Remove the pan from the oven and allow it to cool completely in the pan on a wire rack.
Once completely cooled, carefully lift the square out of the pan using the parchment paper overhang. Transfer it to a cutting board and cut it into squares or desired shapes.
If desired, dust the raspberry lemon squares with powdered sugar for a decorative touch.
Serve the Raspberry Lemon Squares as a delightful dessert or snack. They can be stored in an airtight container in the refrigerator for up to 3-4 days.