To Prepare Dough:
In large bowl, combine flour, sugar and salt. Add butter and then cut in with pastry blender or potato masher, until mixture becomes coarse crumbs the size of peas. In measuring cup, whisk together egg and milk and pour into dry ingredients. Stir with rubber spatula until dough comes together.
Divide in half and wrap each half in plastic wrap, shaping into disks. Refrigerate for at least 1 hour or overnight. Let thaw for 10 minutes at room temperature before rolling if chilled overnight.
To Prepare Filling:
Set large skillet over medium heat and melt butter. Once melted, add crumbled breakfast sausage and cook until browned. Stir in potatoes and cook for about 5 minutes.
In medium bowl, whisk together 3 eggs with 1/2 teaspoon salt and 1/2 teaspoon black pepper. Add eggs to hot skillet and cook until softly scrambled. Remove from heat and let cool to room temperature.
On lightly floured work surface, roll out half of chilled dough into large rectangle about 1/4-inch thick. Trim edges and cut into 8 even rectangles about 3 by 4 inches. Transfer to baking sheet lined with parchment paper. Place in fridge and repeat with second half of dough.
In small bowl, whisk together remaining egg with splash of water. Preheat oven to 375°F.
Add scoop or two of filling onto half of the dough rectangles and sprinkle with bit of cheese. Brush edges with egg wash and then place a second dough rectangle on top of the filling. Crimp edges with fork. Poke a hole or two in the center of each pop tart to allow steam to escape. Brush with egg wash and sprinkle with salt and pepper. Repeat with rest of dough squares and filling.
Bake until golden brown, about 18 to 20 minutes. Remove from heat and let cool slightly before serving. They can also be frozen in freezer storage bag and reheated later as needed.