Preheat the oven to 350 degrees F. Oil a 17-by-12-inch rimmed baking sheet. Line with parchment and oil the parchment. Sprinkle the parchment evenly with 1 1/4 cups of the coconut.
In a medium bowl, whisk together the flour, cornstarch, salt, and the remaining 1/2 cup of coconut. In a large bowl, with an electric mixer on high speed, beat the egg yolks and 1/2 cup of the granulated sugar until opaque, about 1 minute. Add the coconut milk and vanilla on medium speed, then increase the speed to high and continue to beat until the mixture is pale yellow, fluffy, and about doubled in size, 6 to 8 minutes. Set aside.
With clean beaters, in a clean bowl, beat the egg whites on high speed until they have lost that yellow hue, about 1 minute. While beating, pour in the remaining 1/3 cup of sugar. Continue to beat until you have stiff, shiny peaks, 4 to 5 minutes.
With a large rubber spatula, fold the flour mixture into the egg yolk mixture. Then, carefully fold the egg white mixture into the egg yolk mixture, taking care not to deflate it. Dollop the batter onto the prepared pan and then use a long, offset spatula to spread it out evenly. Again, try not to deflate the batter too much. (Without leavener, the whipped eggs are what lift the cake.)
Bake the cake until light golden brown and set, 20 to 24 minutes. Meanwhile, set a clean dish towel on a work surface and sprinkle it evenly with confectioners' sugar. Gently press the center of the cake and it should spring back. Cut around the edges of the warm cake with a sharp knife to release it from the pan, then invert it onto the prepared dish towel. Remove the parchment paper. Roll up the towel and the cake, starting with the short end, and transfer it to a wire rack to cool completely.
To finish: Toss the berries with the granulated sugar and refrigerate until ready to serve, at least 1 hour.
Whip the cream and confectioners' sugar to stiff peaks. (Do not overwhip.) Reserve 2 cups of the whipped cream for serving. Unroll the cake. Spread with the remaining whipped cream and top evenly with the berries, lifting them out of the syrup with a slotted spoon. Roll the cake back up and transfer to a serving plate. To serve, sprinkle the cake with confectioners' sugar before slicing. Serve each piece with extra berries, a drizzle of the reserved berry syrup and dollop of the reserved whipped cream.