We start by removing the puff pastry from the fridge and pre-heating the oven to 375 degrees. Next, we put the butter in a small prep bowl and microwave it on medium power for 30 seconds. We set this aside to cool.
On a lightly floured board, we roll out the puff pastry to fit our 14 × 4 inch rectangular tart pan with removable bottom. The puff pastry sheet was folded in thirds in the package. When we unfold it, we break off two-thirds of the sheet and refrigerate/freeze the rest for another time. We roll a third of the dough at a time. The puff pastry is rectangular, so we just make it a little thinner. We fit both pieces into the tart pan, pressing down the two pieces in the center to join. We make sure the dough is all the way up the sides of the pan. We roll any overhang over with the rolling pin and use to fill-in any thin areas. We put the tart pan into the fridge while we make the filling.
On our cutting board, we coarsely chop the pecans. In the bowl of our KitchenAid stand-up mixer, we combine the sugar and corn syrup. With the paddle attachment, we begin to combine at stir speed.
We break an egg into a small cup and then add to the bowl of the mixer. We raise the speed to 2 and combine. We repeat with the second egg. We stop the mixer after the eggs are combined and, with our spatula, we make sure the corn syrup is not stuck on the bottom of the bowl. We add the cooled, melted butter and put the mixer on speed 4 for 20 seconds. We lower the speed to stir speed, add the vanilla and pecans and mix well. We turn off the mixer and get the apples ready.
We rinse and pat dry our apples. We cut a slice off the top and bottom of the apples. With our peeler, we peel each apple to remove all the skin. Next, we cut the apples in half and then each half into quarters. We remove the small piece of core from each quarter and discard. We chop the apples and put them in a small bowl. We sprinkle the apples with the lemon juice and toss with our clean hands. We then remove the tart pan from the fridge.
We spread the apples in the tart pan. We give the pecan mixture a quick stir with the spatula and then cover the apples with it. We place the pan on a cookie sheet (the mixture may bubble over) and place in the oven. We check after 50 minutes of baking. We want the tart to be just set. It takes about 1 hour and 10 minutes this time.
When the tart is cooked, we remove it from the oven and allow to cool. We refrigerate it covered with plastic wrap. When it is time to slice, we use a butter knife to go around the edges of the pan to loosen it and then remove the tart out of the outer metal ring. We slice and serve.