In a nonstick saucepan, over medium heat, combine the milk, sugar, and vanilla. Whisk to dissolve the sugar. When the mixture comes to a gentle boil, take 1 cup of the milk-sugar mixture and add it to the yolks. Whisk to blend well. Temper the egg mixture to the milk mixture. Cook over medium heat until it thickens slightly, 4 to 5 minutes, whisking occasionally. Dissolve the cornstarch in the water. Over medium heat, slowly add this mixture to the saucepan, whisking constantly for 1 minute. Using a wooden spoon, continue stirring for about 2 minutes. Add the coconut and continue stirring for 2 more minutes. Add the butter and stir until it is completely melted and mixture has thickened to a custard, about 2 minutes. Soften the gelatin in the water and stir into the hot custard. Pour the mixture into a glass bowl. Cover with plastic wrap, pressing the wrap down the surface of the custard to prevent a skin from forming.
Cool completely, stirring occasionally and chill for at least 4 hours. Carefully peel the cake back from the paper. Place the cake on a fresh piece of parchment paper. Roll and unroll the cake several times. Using a spatula, spread the jam, evenly, over the sponge cake. Using the paper underneath to help you, carefully roll the cake tightly. Refrigerate until set, about 1 hour. Remove from the refrigerator and slice into 1/4-inch round slices. Butter a shallow 9-inch steel ring and a baking sheet.
Line both the sides and bottom of the ring with the cake slices, packing them tightly without overlapping them. Fold the whipping cream into the coconut mixture. Spoon the coconut mixture into the mold. Cover with plastic wrap and refrigerate until set, about 3 hours. Remove from the refrigerator and invert onto a glass serving platter. Slice the mold into individual servings and garnish with sweetened whipped cream and mint.