In a medium bowl, whisk together all of the dry ingredients: all-purpose flour, baking soda, cornstarch, and salt.
In a stand mixer using the paddle attachment (can alternatively use hand mixer or whisk), mix the butter, ube halaya, and sugars on medium speed until combined.
Scrape bottom of bowl. Add ube extract and egg, then mix again.
Fold in all of the dry ingredients using a rubber spatula. Cover the dough and chill in the fridge for at least 30 minutes. This reduces the cookies from spreading and enhances the flavor.
Preheat oven to 350°F/177°C and line two baking trays with parchment paper or silicone mats.
Use large cookie dough scoop to gather dough (about 3 tablespoon each). Shape dough into ball, then place onto lined baking tray. Slightly flatten with palm of your hand or bottom of a cup.
Bake for 11-13 minutes. Cookies should look slightly puffy and soft in the center with firm, set edges. They will not brown because of the purple color.
Let cool on pan for about 15 minutes. Cooling the cookies before adding the marshmallows helps the marshmallows keep their shape and not completely melt into a puddle due to the heat.
Once the cookies are mostly cooled, turn on your broiler (can alternatively use a kitchen blowtorch). Place one ube marshmallow on top of each cookie, then broil/blowtorch until the marshmallow is roasted to your liking. If using a broiler, this will take about 1 minute. Keep an eye on them and be careful to not burn them. Remove cookies from broiler and sprinkle flaky sea salt on top.
Let marshmallow cool for 10-15 minutes. If eaten ASAP, the marshmallow will be a gooey mess, unless you prefer it that way. Cooling allows the ube marshmallow to set and have a bit of a stretch when you pull it apart, like in my video.