To "Soft Fry" tortillas: I keep a small dish of oil next to the heated griddle. Paint a little oil onto the pan, and add a tortilla. Fry until poofed and very lightly golden browned, turn, repeat, then place between paper towels to keep soft, warm and pliable. Continue until all are done.
Though I used flour tortillas for this, you could certainly also use soft white corn tortillas if you prefer. (I LOVE them, but the hubs, not so much). I'll work on him. ;)