Although these savory turnovers are made with leftover pot roast, any number of fillings are possible. Try leftover baked ham with prunes, grated Gruyere cheese, and mustard, or combine leftover roasted turkey, roasted squash, and cranberry sauce. If you have no leftovers, use cooked pork or turkey breakfast sausage, thinly sliced Granny Smith apples, and cheddar cheese.
Pastry Dough: In a large mixing bowl, sift together flour and salt. Set aside 1 cup of flour mixture in a small bowl. Using a pastry blender or two forks, cut cold shortening into the large bowl of flour until mixture resembles coarse crumbs. Add broth and vinegar to the small bowl of flour; stir to make a paste. Add the paste to the large bowl; stir with a fork until dough comes together in a ball. Turn dough out onto a floured board, flatten the ball a little, and cut dough into 8 wedges. Shape each wedge into a ball, flatten slightly, and wrap individually with plastic. Chill in refrigerator at least 1 hour or up to 3 days. Alternatively, the dough may be frozen at this point and thawed before rolling.
To assemble: Place a metal cookie sheet or jelly-roll pan in refrigerator. Remove one piece of dough from the refrigerator. Lightly dust counter with flour and cornmeal. Place dough on counter, cover with the plastic it was wrapped in, and roll dough into a 9- to 10-inch circle, about ⅛ inch thick. Brush a one-inch border of egg white onto the bottom half of the circle. Spread 1 tablespoon chutney on bottom half of the dough, keeping clear of the egg white. Cover chutney with a heaping ½ cup of filling. Fold top half of dough over filling, bringing edges together to form a half-moon shape. Pinch edges of dough to seal, brush border with egg white, then fold border toward filling and pinch again. Place on cookie sheet in refrigerator. Repeat with remaining dough to form 8 pasties. Refrigerate pasties at least 20 minutes, or freeze.
Preheat oven to 425 degrees F. If you have a pizza stone, place it on a rack in the middle of the oven. Place sheet of pasties on top of pizza stone or on rack. Bake 30 minutes for refrigerated pasties, 40 minutes for frozen pasties, until crust tops and bottoms are golden brown. Cover with a clean tea towel and keep warm until ready to serve.