Heat the frozen mangoes in a saucepan over medium heat until softened, about 5-7 minutes. Mash the mangoes into a smooth consistency, then stir in white sugar until fully dissolved. Let cool and refrigerate.

Soak the sweet rice in water for at least 4 hours or overnight. Drain and steam in a lined steamer for about 25-30 minutes until sticky and translucent. Mix with coconut milk and sweetened condensed milk.

Cream together unsalted butter, light brown sugar, and white sugar until fluffy, about 3-4 minutes. Add egg yolk and thick egg white, then mix in vanilla extract, baking soda, and salt. Gradually add flour until just combined. Chill for 30 minutes.

Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper. Scoop about a tablespoon of dough, flatten it, place a dollop of sticky rice mixture and a spoonful of mango jam in the center, folding the dough over. Bake for 13-15 minutes until edges are golden.

Cool cookies on the baking sheet for 5 minutes before transferring to a wire rack. Cookies should have a soft, chewy center with slightly crisp edges.