Preheat oven to 375 degrees. Press the pie crust into a 9 inch tart pan covering the bottom and side. Cut off excess crust with a sharp knife. Line the pecan in a single layer on the bottom of the crust in circles, getting smaller as you go to the center. Refrigerate while you make the filling.
In a medium, heavy bottomed saucepan add the brown sugar, Karo syrup, and butter. Bring to a boil over medium heat, stirring gently. Remove from heat. Using a ladle, remove a small amount of the hot mixture. Slowly pour the hot mixture into the beaten eggs, whisking constantly. Repeat a couple of times until half the hot mixture and the eggs are combined without scrambling the eggs. Add this to the remaining hot mixture and return to a boil. Boil gently for two minutes, whisking gently. Remove from heat, stir in vanilla and allow to cool slightly.
Remove the prepared shell from the refrigerator place it an a baking sheet. Slowly pour the still hot, but not boiling filling over the top of the pecans. Hint: Use a heat-proof measuring cup with a spout and pour slowly. The pecans will float to the top, but if you pour very slowly they will stay in the circle pattern. Carefully place the baking try onto the center rack of the oven. Bake for 20--25 minutes until the filling just jiggles slightly when you move the pan. Remove from the oven, cool and completely on the baking tray. Refrigerate overnight or at least a few hours until the filling is completely set. Cut into small wedges for serving.