Preheat the oven to 400 F (200 C). Grease a 10½ × 15½-inch (27 cm × 39 cm) jelly roll pan with cooking spray and line the bottom with parchment paper.
To the bowl of a stand mixer fitted with the whisk, add eggs, granulated sugar, and salt. Beat on medium-high speed until thick, pale, and ribbony, 5 to 7 minutes. Beat in the vanilla.
Sift in the flour and baking powder and fold until just combined.
Scrape the batter into the prepared pan and spread into an even layer. Bake until golden and a toothpick inserted into the center of the cake comes out clean, about 10 minutes.
Meanwhile, place a clean tea towel on your work surface and sift a generous dusting of powdered sugar all over it.
Once the cake is finished baking, swiftly invert the pan onto the prepared tea towel and carefully remove the parchment. Fold about 3 inches (7.5 cm) of the tea towel over one of the short sides and position the cake so this side is closest to you. Tightly, but gently, roll the cake (with the tea towel tucked inside) away from you. Let cool completely, about 40 minutes.
For the Filling
Using a spice grinder, grind the freeze-dried mango into a fine powder.
In the bowl of a stand mixer fitted with the whisk, beat the heavy cream, granulated sugar, and salt on medium speed until soft and sturdy peaks form, about 2 minutes. Transfer 1 cup (100 g) to a small bowl and set aside in the refrigerator (this is for the garnish).
Add the mango powder to the rest of the whipped cream in the stand mixer bowl and continue to beat on medium speed until stiff peaks form, about 30 seconds (it should have a little bit of a frosting look to it).
To Assemble
Gently unroll the sponge. Gradually roll it back up (without the towel this time) and use a small paring knife to scrape off the visible golden “crust” from the exterior of the cake (it should peel right off, creating a flawless finish).
Unroll the sponge onto a sheet of parchment paper. Top with the jam and spread evenly. Top with the mango cream and spread evenly.
Starting with the same short side you started with before, roll up the cake. Using a serrated knife, slice a bit off from both ends.
Swiftly transfer the cake to a serving platter. Dollop the reserved whipped cream along the top of the cake and spread, swooping and swirling, to create soft whimsical peaks. Garnish with diced mangoes and sliced strawberries.