In a large bowl, whisk together the gluten free flour, tapioca starch, sugar, xanthan gum, and salt. Add the cold cubed butter and cream cheese, then use a pastry cutter or your fingertips to work them in until the mixture resembles coarse crumbs with some pea-sized butter pieces.Add the egg and ice water, then stir until the dough comes together. Shape it into a flat disc, wrap in plastic wrap, and refrigerate for at least 1 hour.
While the dough chills, make the filling. In a medium saucepan over medium heat, combine the diced strawberries, sugar, cornstarch, and lemon juice. Cook, stirring frequently, for 5 to 7 minutes until the strawberries break down and the mixture thickens into a jammy consistency. Remove from heat and let cool completely.
Preheat the oven to 400 degrees F and line a baking sheet with parchment paper. Roll the chilled dough out on a well-floured surface to about 1/8 inch thickness. Cut out 6 squares or rectangles, roughly 5 inches each. You may need to re-roll scraps.
Spoon about 2 tablespoons of the cooled strawberry filling onto one half of each dough square, leaving a 1/2 inch border. Fold the dough over diagonally to form a triangle or fold in half for a rectangle shape. Press the edges firmly with a fork to seal.
Place the turnovers on the prepared baking sheet. Brush each one with beaten egg. Use a fork to poke 2 to 3 small vent holes in the top of each turnover.Bake for 20 to 25 minutes, until puffed and deeply golden brown.
While the turnovers cool slightly, whisk together the powdered sugar, milk, and vanilla until a smooth glaze forms. Drizzle generously over the warm turnovers and let the glaze set for a few minutes before serving.