Position a rack in the center of the oven and heat to 325°F (160°C). Line a 9x13-inch baking pan with parchment paper, leaving a 2-inch overhang on the long sides for easy removal.
Place the raspberries in a blender or food processor and pulse until smooth. Pour the puree through a fine-mesh sieve into a small saucepan, pressing with a spatula to remove seeds. Discard seeds.
Simmer the seedless puree over medium-low heat, stirring occasionally, until it thickens and reduces to about 1/3 cup (15–20 minutes). Transfer to a bowl and let cool while you prepare the crust.
In a medium bowl whisk together the flour, 1/2 cup sugar, 1 tablespoon cornstarch and the salt so the dry ingredients are evenly distributed.
Stir the melted butter into the dry mix until a crumbly dough forms. It should hold together when pressed; if it's too dry add a teaspoon of cold water at a time.
Press the dough into the prepared pan, working it up slightly along the edges to form a low rim. Use the bottom of a measuring cup for an even surface. Bake for 18–22 minutes, until the crust is set and just beginning to color.
Remove the crust from the oven and let it cool 8–10 minutes. Using a fork, gently poke the surface a few times (do not pierce through to the bottom) to help the filling adhere without draining.
While the crust bakes, whisk together 1 1/3 cups sugar and 1/3 cup cornstarch in a large bowl until lump-free. Add the eggs one at a time, whisking thoroughly after each addition.
Stir the cooled reduced raspberry puree into the egg mixture, followed by the lemon juice and lemon zest. Whisk until smooth — the mixture will be fairly loose but homogenous.
Carefully pour the lemon-raspberry custard over the warm shortbread crust and return the pan to the oven. Bake 22–28 minutes, until the filling is set at the edges and only slightly jiggly in the center.
Let the bars cool at room temperature for 1 hour, then cover and chill in the refrigerator at least 3 hours (overnight is best) to firm before slicing.
Lift the bars from the pan using the parchment overhang and place on a cutting board. Use a sharp knife wiped between cuts to slice into 16 bars for clean edges. Store in an airtight container in the fridge for up to 4 days.