Flip the cake onto the towel (parchment side up), then carefully peel away the parchment paper. Starting from the short end, roll the cake up with the towel inside, forming a loose log. Rolling it while warm helps the cake retain its shape and prevents cracks. Let the rolled cake cool completely, seam-side down, on a wire rack.
While the cake cools, make the yoghurt filling. In a large chilled bowl, whip 1/2 cup heavy cream using an electric mixer on medium-high speed until soft peaks form. Gradually add the 2 tbsp powdered sugar and continue whipping until stiff peaks form. In a separate bowl, mix 1 cup Greek yoghurt with 1 tsp vanilla extract.
