Prepare Mango Jam: Place the mangoes in a saucepan over medium heat for about 5-7 minutes, stirring occasionally until softened. Mash gently with a fork, mixing in sugar to taste. Cool in the refrigerator for at least 30 minutes.

Cook Sticky Rice: Rinse sweet rice under cold water and soak for at least 4 hours or overnight. Steam the rice in a lined steamer for about 20 minutes. Combine with coconut milk and sweetened condensed milk.

Make Cookie Dough: Cream together softened butter, light brown sugar, and white sugar until fluffy. Incorporate egg yolk, thick egg white, and vanilla extract. Gradually mix in baking soda, salt, and flour until just combined. Chill for at least 30 minutes.

Assemble and Bake: Preheat oven to 350°F (175°C). Line a baking tray with parchment paper. Flatten spoonfuls of dough, dollop with mango jam and reserved sticky rice. Bake for 13-15 minutes until edges are golden.

Cool and Serve: Let the cookies cool on the tray for about 10 minutes, then transfer to a wire rack. Serve warm or at room temperature.