Assuming your cake came out well and not overly wet (though I appreciate this is a particularly moist cake/batter), I'm going to guess that this was just because of poor airflow when cooling.
Did you cool the whole pan on a wire rack as suggested? If not, that would be the first thing I tried to fix this. This could help the cake cool faster, allowing less opportunity for condensation.
But some other things to try could include:
- Taking the cake out of the pan as soon as it's set enough to move, and letting it cool further on a wire rack. I appreciate part of this cake's appeal is how beautifully soft it looks though, so I imagine this may not be an easy thing to do!
- If you baked in a springform pan, could you take the sides off the pan sooner to allow for better airflow?
- The third thing is I wonder whether the double layer of baking paper is having an impact here? This may be hurting airflow even further and trapping more moisture.
Just some ideas! Besides a batter being too wet, poor airflow seemed to be the main culprit of condensation building up on the bottom of a cake that I could find, so seems like a good place to start troubleshooting. Good luck next time - the recipe looks delicious.