Make the filling: In a saucepan over medium heat, combine strawberries, sugar, cornstarch, lemon juice, and vanilla. Cook for about 5–7 minutes, stirring often, until thickened. Let cool completely.

Make the crust: In a large bowl, mix flour, sugar, and salt. Add cold butter and use a pastry cutter or your hands to cut it in until the mixture looks like coarse crumbs. Add ice water a tablespoon at a time until the dough comes together. Divide dough in half, wrap in plastic, and chill for at least 30 minutes.

Roll and fill: Preheat oven to 375°F (190°C). Roll out one dough half on a floured surface to about 1/8-inch thick. Cut into 4-inch circles or squares. Add a tablespoon of strawberry filling to the center of half the pieces. Top with another piece of dough and crimp the edges with a fork to seal.

Egg wash & bake: Place hand pies on a parchment-lined baking sheet. Beat the egg with water and brush over the tops. Sprinkle with coarse sugar if using. Cut a small slit in the top of each pie for steam to escape.

Bake: Bake for 20–25 minutes, or until golden brown. Let cool slightly before serving.