Mix 3 cups flour with salt. With a pastry blender, cut in shortening until crumbly. With a fork, gradually mix in just enough water (about 1/2 cup) until dough holds together. Divide dough into 3 equal portions.
On a floured board, roll out each portion into an oval 1/8 inch thick and 8 to 10 inches long. Cut lengthwise into 1-inch-wide strips.
In an ovenproof 5- to 6-quart pan that's about 10 inches wide, combine vinegar, sugar, vanilla, and 4 cups water. Bring to a boil over high heat.
Reserve 8 pastry strips, and tear the remainder into pieces about 1/2 inch long. Drop torn pieces into boiling liquid. Boil 5 minutes.
Remove pan from heat. Crisscross remaining 8 strips on top of the cooked dough pieces, making a lattice and trimming ends to fit. Push leftover trimmed dough scraps into syrup.
Bake in a 425° oven until lattice top is golden brown, 25 to 30 minutes. Serve hot or warm. Spoon into bowls.