Combine flour and shortening in a large bowl. Work the shortening into the flour with a pastry blender or your fingers until completely incorporated.
Dissolve the salt in the water and pour ⅔ cup of the salt water over top of the flour mixture. Immediately work it in with a fork. The dough will be in large clumps.
Scoop the soft dough onto your work surface and knead until smooth.Divide dough into 12 equal pieces and roll each piece of dough into a smooth ball. Set the tortilla dough balls on a plate, cover with sheets of plastic wrap or damp cloth or paper towel and let dough rest for 30 minutes at room temperature.
After the resting period, heat a cast-iron skillet or a large frying pan over medium-high heat. On a lightly floured surface, roll out a portion of the dough into a 7-inch circle. Flatten a ball of dough with the palm of your hand, sprinkle with a little extra flour, then roll forward and back across it, rotating a sixth of a turn dough and roll forward and back again. Continue until it is a 7-inch circle. Dust tortilla with flour as needed while rolling.
If using a tortilla press: Using two pieces of parchment paper, place one piece of paper on the press, place the dough ball in the center and top with another piece of parchment. Then press. Lift the tortilla press, and flip the tortilla over (still sandwiched between the parchment. Press firmly one more time. Then remove the tortilla and proceed to cook.
To cook, lay the first tortilla on the hot griddle. After 30-45 seconds, when air bubbles pop up and golden brown spots show underneath, flip it over. cook another 30-45 seconds more, until the second side is browned. Do not overcook or the tortilla will become crisp.
Remove and wrap in a clean kitchen towel. Repeat procedure with remaining tortillas and stack them on top of each other.