Add the 2 ½ cups all-purpose flour, 1 tablespoon granulated white sugar, 1 tablespoon baking powder, 1 teaspoon kosher salt, ½ teaspoon baking soda, and ½ teaspoon ground mustard powder to the bowl of your food processor.
Cut the 8 tablespoon unsalted butter into 4 pieces and add it to the food processor. Pulse a few times to chop and incorporate the butter into the dry ingredients. Continue processing until the butter pieces are the size of peas.
Fold the 1 cup cubed ham, ½ cup cheese , ½ cup chopped green onions, and 2 cloves of garlic into the dry ingredients.
Add the 1 cup sourdough discard, 1 large egg, and ½ teaspoon white vinegar to the dry ingredients and stir to combine.
Slowly stir in the ⅔ cup heavy cream. The amount of cream will vary based on how liquid or thick your sourdough starter is. Drizzle in just enough cream until the dough becomes cohesive.
Squeeze the dough, when it holds together with no crumbs left in the bottom of the bowl, you've added enough cream.
Pour the dough onto a lightly floured surface. Knead the dough a few times until it begins to come together. Flatten the dough and fold it into thirds like you would a letter. Give the dough a quarter turn and repeat this flattening and folding process two more times.
Pat into a disc approximately 1 inch thick and cut into eight wedges using a sharp knife or use a circle biscuit cutter to cut out about 10 scones.
Transfer the scones to a parchment paper-lined baking sheet. To prevent the scones from spreading chill in the refrigerator for 30 minutes.
Preheat the oven to 425°F (220°C).
Brush the scones with heavy cream to encourage browning.
Bake for 22-25 minutes until golden brown.