In a food processor, pulse together flour, salt and sugar. Add butter and pulse until combined (you should still see pea sized pieces of butter). Start with ½ cup of sourdough discard. Pour over flour mixture and pulse until dough comes together. If dough isn't coming together, add more discard until it does. Press dough into a disc, cover and refrigerate for at least one hour.
Dice cheese into 1 inch cubes.
Preheat oven to 375. Atop a silicone baking mat or parchment paper, roll dough into a large circle. Pile cheese on half of the circle. Fold the dough over top and seal the edge with a fork. Sprinkle with sugar.
Bake for 30 minutes, or until golden brown.
Notes
If you can't find oaxaca, a fresh, low moisture mozzarella is a great substitute.