Prep: Preheat oven to 400°F (204°C). Prick thawed pie crust all over with a fork, then bake it (empty) for 15 minutes, or until it begins to brown. When finished, reduce oven temperature to 350°F (176°C).
Clean Leeks: Meanwhile, cut the root and dark green part off of each leek, then cut each in half lengthwise. Slice into ¼ inch thick half moons. Transfer to a large bowl, then cover with water. Swish around to clean out the dirt, then drain.
Saute Leeks: Heat butter or oil in a large saute pan over medium/high heat. Add leeks and salt, then cook until tender and bright green, 7 to 10 minutes.
Assemble: Spoon leeks into pie shell (use as much of them as you can, but ensure the pie is only about ¾ filled to the top). Break up the goat cheese and dollop all over the leeks. In a separate bowl, whisk together egg and half & half until well blended, then pour it over the leeks to fill the pie.
Bake: Bake for about 45 minutes, or until the outside is completely set and the middle still jiggle a touch. If needed, cover with aluminum foil to prevent the crust from over browning.