In a medium saucepan over medium heat, melt ¾ cup of sugar until deep golden brown, swirling the pan gently for even cooking (about 8-10 minutes). Pour melted caramel into a greased 9-inch round cake pan and set aside to cool.

In a large mixing bowl, whisk together 10 egg yolks, 1 can of sweetened condensed milk, 1 can of evaporated milk, and 1 teaspoon of vanilla extract until smooth. Strain this mixture through a fine-mesh sieve into the cooled caramel-lined cake pan.

In another bowl, whisk together 1 cup of cake flour, 1½ teaspoons of baking powder, and ¼ teaspoon of salt until combined.

In a separate large bowl, combine 10 egg yolks, ¼ cup of sugar, ¼ cup of vegetable oil, ⅓ cup of ube halaya, optional ½ teaspoon of ube extract, and ⅓ cup of water. Whisk until smooth, then gently fold in the dry ingredient mixture.

In a clean mixing bowl, beat 10 egg whites with a pinch of cream of tartar until foamy. Gradually add ¼ cup sugar, increasing the speed to high, and beat until stiff peaks form.

Fold the meringue into the ube batter in three increments. Pour the ube chiffon batter gently over the flan mixture in the cake pan. Place the cake pan in a larger roasting pan filled with 1 inch of hot water. Bake at 350°F (175°C) for 1 hour, or until a toothpick comes out clean.

Cool the cake in the pan for about 10 minutes, then refrigerate for at least 4 hours before serving. To serve, run a knife around the edges and invert onto a serving plate.