In a stand mixer, or by hand, stir flour and 1 teaspoon salt together, then beat in butter until the flour resembles corn meal. Beat in cheddar cheese and sourdough starter. Slowly add cold water, bit by bit, until the dough forms a ball and becomes pliable.

Wrap the ball of dough in plastic wrap and refrigerate for about 8 hours, then remove from fridge and allow it to warm at room temperature for about 20 minutes. Cut the dough into four equal quarters. Place each ball of dough between 2 sheets of parchment paper and roll it out until it forms a circle about ⅛-inch thick.

Melt the butter in a large skillet over medium heat. When the butter froths, stir in vegetables, remaining teaspoon salt, and cook them gently for 6 to 8 minutes or until they release their fragrance. Stir in chicken and broth. Simmer for a further 15 to 20 minutes or until liquid has largely evaporated.

Take a disc of dough, and spoon approximately ¼ of the chicken and vegetable filling and 2 tablespoons cream cheese onto ½ half of the disc. Fold the other half of the dough over the filling and seal with a fork or by crimping the edges. Continue working until all 4 turnovers have been filled.

Whisk beaten egg with a tablespoon or two of water to make an egg wash, then brush each pastry with egg wash.

Cut a small slit in the top crust of each turnover to allow for escaping steam, then bake at 375 F about 45 minutes or until the crust is golden brown. Allow turnovers to cool on a wire rack for 5 minutes before serving.