If using Greek yogurt, you can skip this step and go straight to next step. If not using Greek yogurt, place the yogurt in a cheesecloth-lined sieve set over a medium bowl. Let drain at least 30 minutes or cover and refrigerate overnight. Discard the liquid.
With an electric mixer on medium-low speed, beat the cream cheese until very smooth. Add the sugar and vanilla and beat until smooth. Reduce the speed to low, add the yogurt, and beat until smooth.
Pour the filling into the cooled crust, cover loosely, and refrigerate until set, at least 6 hours or up to 1 day.