In a large skillet over medium heat, brown the breakfast sausage until fully cooked, about 5-7 minutes. Stir in flour, then whisk in whole milk, cooking until thickened, about 3-5 minutes. Season with salt and pepper, then let cool.

In a separate skillet, whisk 4 eggs over medium-low heat until softly scrambled, about 4-5 minutes. Remove from heat when still slightly runny and set aside to cool.

Preheat oven to 375°F and line a baking sheet with parchment. In a mixing bowl, combine flour and salt. Cut in shortening and butter until coarse crumbs form, about 3-4 minutes.

Whisk together the egg, vinegar, and cold water. Pour into the flour mixture and stir until dough forms. Wrap in plastic and refrigerate for 30 minutes.

Roll out the chilled dough to 1/8-inch thickness. Cut into 3x4 inch rectangles, yielding about 8. Place half on the baking sheet.

Spread sausage gravy on uncovered rectangles, top with scrambled eggs and shredded cheese if desired. Place remaining rectangles on top and crimp edges to seal.

Brush the tops with beaten egg for a golden finish. Bake for 25 minutes until golden brown and puffed. Keep an eye on them to prevent over-baking.

Allow to cool on the baking sheet for 5 minutes, then transfer to a wire rack. Serve warm or store in an airtight container.