In the bowl of a food processor, pulse flour, 2 teaspoons sugar, and 1 teaspoon salt until well-combined. Add cold butter cubes and shortening; pulse mixture repeatedly until butter and shortening are incorporated and butter is the size of small peas. Gradually pour in ½ cup ice water (ice removed!), pulsing mixture repeatedly, just until a dough comes together (if the dough seems dry, add in 1 to 2 tablespoons more ice water).

Divide dough in half; transfer each piece to a sheet of plastic wrap. Gently pat each piece of dough into a disc shape, about 1 inch thick, and cover tightly with plastic wrap. Refrigerate for at least 2 hours, or up to 3 days, before using.

In a medium bowl, toss berries with ½ cup sugar, cornstarch, lemon juice, 1 teaspoon vanilla extract, cinnamon, and ½ teaspoon salt until well-combined. Allow to sit for 15 minutes so flavors can meld and berries begin releasing their juices. Transfer the mixture to a medium saucepan; heat over medium-high heat for 5 minutes or until the mixture begins to boil. Once boiling, use a potato masher or the back of a large spoon to mash berries as much as possible. Reduce heat to medium; cook berry mixture, stirring constantly, for 7 to 9 minutes more or until the mixture is thickened and resembles a loose jam. Transfer mixture to a clean medium bowl; set aside to cool completely at room temperature (mixture will thicken as it stands). Once cooled, cover and transfer the bowl to the fridge to chill until your pastry crust is ready.

Line one large baking sheet with parchment paper. Set aside. Remove one dough disc from the fridge. On a lightly floured surface, use a rolling pin to roll out dough to a rough 9x12-inch rectangle, about ⅛-inch thick. Use a knife, pizza cutter or bench scraper to trim edges to form a perfect 9x12-inch rectangle. Cut dough into nine smaller rectangles, each about 3x4 inches. Transfer rectangles to lined baking sheet, spaced at least 1 inch apart. Transfer baking sheet to the fridge while you prepare the second dough disc.

Using second dough disc, repeat the rolling and cutting. Remove baking sheet and cooled berry filling from the fridge. Brush each rectangle on the baking sheet with egg wash, then top each rectangle with approximately 1 heaping tablespoon berry filling, spreading to within ½-inch of the edges. Top each with remaining 9 rectangles to form tarts. Use fork tines to seal the edges, then poke a few holes on the tops of each tart. Brush the tops of each tart with more egg wash, then transfer the baking sheet to the freezer for 15 minutes to chill and set the tarts.

Heat oven to 375°F. Once tarts are fully chilled, brush the tops again with egg wash. Bake for 23 to 27 minutes, rotating once halfway through baking, until pastry is puffed and golden. Cool tarts completely on the baking sheet.

Press 1 ½ tablespoons berry filling through a fine-mesh strainer into a medium bowl to remove seeds and bits (you should end up with about 1 tablespoon jam in the bowl). Add powdered sugar, milk, and ½ teaspoon vanilla extract; whisk until a smooth, thick icing forms.

Spoon and spread icing evenly over fully cooled tarts. Top with sprinkles, if desired. Allow icing to set completely before serving, about 20 to 30 minutes.