For the crust:

1. Preheat oven to 325° F. Line an 8-inch square baking pan with parchment paper, leaving a two-inch overhang. Brush the parchment paper and sides of the pan with softened butter.
2. Using a medium bowl, combine the sugar, flour, baking powder, salt and cinnamon. Add the butter and vanilla, tossing to coat in flour mixture. Using two dinner knives or a pastry dough blender, cut the butter into the flour mixture until no large pieces remain and the butter is the size of a pea. Transfer dough to prepared pan, using your hands to flatten dough into an even layer. Refrigerate and chill for 30 minutes. 
3. Bake the crust until golden brown around the edges, about 25 minutes, set aside to cool.

For the filling:

1. Preheat oven to 325° F. Using a medium saucepan over medium heat, combine lemon juice, salt, blackberries, blueberries, raspberries and ½ cup sugar, stirring to dissolve sugar. Reduce heat to medium low and simmer, stirring occasionally, until berries have collapsed and released their juices, about 10 minutes. Using an immersion blender, blend the berry mixture until smooth. Place a fine-mesh sieve over a small bowl or measuring glass and strain berry puree, pressing on the solids with the back of a spoon. Discard solids, leaving about 2 cups berry puree.
2. Using a small bowl, whisk together cornstarch and remaining cup of sugar, set aside. Using a medium saucepan over low heat, whisk berry puree, egg yolks and egg to combine. Gradually whisk in cornstarch-sugar mixture and cook, whisking constantly, until mixture thickens and wires of the whisk leave a trail when lifted, about 10 minutes. Remove from heat and whisk in lemon zest and butter until incorporated.
3. Pour filling over prepared crust and spread into an even layer. Bake until filling is set, about 25 minutes. Place on a wire track to cool, chill at least 2 hours or overnight in the refrigerator.
4. Run a paring knife around the sides of the pan. Using the parchment paper, lift the bars from the pan and place on a cutting board. Cut bars into 16 squares and garnish with fresh berries.