Step 1: Preheat your oven to 375°F (190°C). Line a 10x15 inch jelly roll pan with parchment paper, extending over the sides, and lightly grease the paper.
Step 2: Prepare the Génoise Sponge: In a large heatproof bowl set over a saucepan of simmering water (bain-marie), combine eggs and granulated sugar. Whisk constantly until the mixture is warm to the touch (about 110°F/43°C) and the sugar has dissolved. Do not let the bowl touch the water.
Step 3: Remove the bowl from heat and beat with an electric mixer on high speed for 8-10 minutes, until very thick, pale, and tripled in volume. The mixture should form a thick ribbon when the beaters are lifted. Gently fold in the vanilla extract.
Step 4: Sift half of the all-purpose flour over the egg mixture and gently fold it in with a rubber spatula until just combined. Repeat with the remaining flour, folding until no streaks remain. Drizzle the melted butter around the edge of the bowl and gently fold it in.
Step 5: Pour the batter evenly into the prepared jelly roll pan and spread gently to the corners. Bake for 10-12 minutes, or until the cake springs back when lightly touched and the edges are lightly golden.
Step 6: While the cake is still hot, invert it onto a clean kitchen towel lightly dusted with powdered sugar. Carefully peel off the parchment paper. Trim any crisp edges if desired. Starting from a short end, carefully roll the cake and towel together. Let it cool completely in this rolled shape on a wire rack (about 30-45 minutes).
Step 7: Prepare the Mango Passion Fruit Cream: In a large bowl, combine chilled heavy cream, powdered sugar, and vanilla extract. Beat with an electric mixer until soft peaks form. Gently fold in the mango puree and passion fruit pulp until just combined. Be careful not to overmix.
Step 8: Assemble the Yule Log: Once the cake is completely cool, carefully unroll it. Spread the mango passion fruit cream evenly over the entire surface of the cake, leaving a small border at one short end. Gently re-roll the cake tightly, without the towel.
Step 9: Carefully transfer the rolled cake to a serving platter, seam-side down. Refrigerate for at least 2 hours, or until firm, to allow the cream to set.
Step 10: Prepare the Tropical Glaze: In a small saucepan, combine mango puree, passion fruit pulp, and powdered sugar. If using, whisk in cornstarch until dissolved. Heat over medium-low heat, stirring constantly, until the mixture thickens slightly and becomes glossy. Let cool slightly.
Step 11: Pour the cooled glaze over the chilled Yule Log, allowing it to drip attractively down the sides. Decorate with fresh mango slices, passion fruit seeds, or mint sprigs as desired. Return to the refrigerator for at least 30 minutes to set the glaze.
Step 12: Slice with a sharp knife and serve chilled. Enjoy your tropical holiday treat!