In a medium bowl, combine the diced strawberries, 1/4 cup granulated sugar, cornstarch, and lemon juice. Toss gently until the strawberries are evenly coated. Set aside to macerate for 15-20 minutes while you prepare the shortcake base. This allows the strawberries to release their juices and the sugar to dissolve.
Preheat your oven to 375°F (190°C). Line an 8x8 inch baking pan with parchment paper, leaving an overhang on two sides for easy removal. In a large bowl, whisk together the flour, 1/2 cup sugar, baking powder, and salt. Using a pastry cutter, two forks, or your fingertips, cut the cold cubed butter into the flour mixture until it resembles coarse crumbs with some pea-sized pieces of butter remaining.
In a small bowl or measuring cup, whisk together 2/3 cup heavy cream, the egg, and 1 tsp vanilla extract. Pour the wet ingredients into the flour-butter mixture. Stir with a fork just until a shaggy dough forms—do not overmix. Press about two-thirds of the dough firmly and evenly into the bottom of the prepared baking pan to form the base layer.
Using a slotted spoon to leave excess liquid behind, spread the macerated strawberry filling evenly over the pressed dough base. Crumble the remaining one-third of the shortcake dough over the strawberry layer in an even, streusel-like topping. Brush the top lightly with the remaining 1 tablespoon of heavy cream.
Bake for 28-32 minutes, or until the top is golden brown and a toothpick inserted into the shortcake layer (not the strawberry center) comes out clean. Place the pan on a wire rack and allow the bars to cool completely in the pan, about 1-2 hours. This is crucial for clean slicing.
While the bars cool, make the icing: In a small bowl, whisk together the sifted powdered sugar, 2 tablespoons of cream or milk, and 1/2 tsp vanilla until smooth. Add more liquid, 1 teaspoon at a time, until you reach a thick but pourable consistency. Once the bars are completely cool, use the parchment paper to lift them from the pan. Drizzle the icing over the top in a zig-zag pattern. Slice into 9 squares and serve.