Before you begin your tart, you need to prepare the apricots. Put the cut apricots in a bowl with one-quarter cup of the sugar. Stir them around so the sugar is coating the apricots. Let this sit on your counter or in the refrigerator for about 2 hours. This process is called macerating. You're getting the apricots nice and sweet, and the extra sugar and juice from the apricots will make a little sauce.
Now take your piecrust out of the refrigerator and let it sit on your counter for about 15 minutes. This will make it easier to unroll.
Roll it out and put it in a 9-inch tart pan. The edges will hang over. Just fold them back inside and crimp so you make a rim on your tart. Bake the tart at 400 degrees for about 15 minutes or until golden brown.
In a medium bowl, put in your cream cheese, the sour cream, the vanilla and one-quarter cup of the sugar. Beat until fluffy. Let sit until the piecrust is baked and it has cooled off.
To assemble your tart, spread the cream cheese mixture in the bottom of the cooled tart. Place the apricots one by one on top of the cream cheese, cut side down. They will look like little orange bubbles on top of the cream cheese. If you want your tart to look fancy and symmetrical, start with one apricot in the middle and then work your way to the outside. Throw away the rest of the juice that has accumulated in the bottom of the bowl. Believe me, this tart will be sweet enough.
Chill the tart until ready to serve. Keep leftovers in the refrigerator. It will stay in pretty good shape for about 3 days.