In a bowl, mix the sourdough starter, flour, salt, and sugar until crumbly.
Add the cold, cubed butter and mix until the mixture resembles coarse crumbs.
Stir in the egg and vanilla extract until a dough forms.
Shape the dough into a rectangle, wrap it in plastic, and refrigerate for at least 30 minutes.
Preheat your oven to 375°F (190°C).
Roll out the chilled dough on a floured surface to about 1/8 inch thick.
Cut the dough into rectangles (about 3x4 inches) for the pop tarts.
Place a spoonful of your chosen filling on half of the rectangles.
Top with remaining rectangles, sealing the edges with a fork.
Brush the tops with beaten egg and cut small slits for steam to escape.
Bake for 20 minutes or until golden brown.
Allow to cool, then drizzle with icing if desired.