Preheat the oven to 350ºF. Line an 8-inch square baking pan with parchment paper, and grease with nonstick cooking spray.
In the bowl of a stand mixer fitted with a paddle attachment, beat the sugar and lemon zest together. Add the butter, and continue beating on high until creamy. Add the vanilla and salt, and continue beating until blended. Slowly mix in the flour.
Press dough into the bottom of the prepared pan and bake for 15-18 minutes, or until lightly browned around the edges.
While crust is baking, prepare the filling. Place raspberries into a food processor and puree until smooth. Pour raspberry puree through a fine mesh strainer set over a bowl, pushing out as much juice as possible. Discard the seeds, and measure out 1/2 cup raspberry juice.
In a large bowl, whisk the eggs, sugar, zest, and salt together in a large bowl. Whisk in the raspberry juice and lemon juice, and then the flour, until combined. Remove the parbaked crust from the oven. Pour the filling onto the warm crust.
Reduce oven temperature to 325ºF, and bake for 22-26 minutes, or until the center is set. Place pan onto a wire rack to cool completely, about 2 hours. Once cool, cover bars and chill in the refrigerator for 2 more hours.
Using a sharp knife dipped in hot water and wiped dry, cut raspberry bars into squares. Dust with powdered sugar just before serving.
Notes
*If using frozen raspberries, make sure to thaw them out before pureeing.