In a food processor, beat sugar and butter together. It should look light and fluffy. Add in vanilla extract.
In another bowl, mix dry ingredients flour, salt and baking powder. Give it stir. Gradually add the dry ingredients in the butter mixture.
Mix until its combined, it might still look dry, add milk. Pulse the food processor few times and remove the mixture in a bowl. Mix and form dough with hands. As I said it might look dry but will come together.
Remove about half cup dough and leave it aside.
Transfer the remaining dough into a parchment paper and form a rough disc or small rectangle. Cover it and chill it for 10 minutes or until its slightly firm.
Meanwhile, in the same food processor. Take black sesame seeds and sugar and pulse it until it becomes paste. It will stick to the edges of the food processor and look sticky. Add in one tbsp of oil and pulse little bit more. Stir in between to make sure it is combining well. In this paste, add the reserved half cup of dough. Pulse/mix to combine and it will form our sesame seed dough.
Bring out the plain dough from freezer, roll out into a rectangle on parchment paper. Roll dough between two parchment paper for easier rolling. Now, place the black sesame dough with your hands evenly, leaving ½ inch on sides. Make sure the dough looks smooth. Use spatula or fingers to press.
Starting on the long side of the rectangle, carefully and gently roll the dough into a tight roll. Try to not leave gap in the middle. Parchment paper will come handy while rolling. If your dough is too soft place it for few minutes in the freezer and if its too hard then leave it for few seconds.
Once the dough is rolled, wrap in same parchment paper, roll few times, smooth out the log by rolling couple of times back and forth and cover it in same parchment paper refrigerate until its firm at least 20-30 minutes or overnight.
When ready to bake, preheat the oven to 350 F. Line the baking sheets with parchment paper.
Unwrap the dough log (if using sanding black sugar, roll the whole log into sugar back and forth until sugar sticks to it and then slice them, if not using the sanding sugar, then proceed with slicing. Using a very sharp knife slice ¼ inch rounds and place them on prepared baking sheets 1 inch apart. They do not spread a lot, but they do spread a little bit. So, you want to make sure they are not touching each other.
Bake the cookies for about 15-17 minutes or until they are cooked through. They will look pale to light pink color specially the plain dough. They will still feel soft though, they firm up once cooled down.
Remove cookies on cooling rack and let them completely cool down.
Serve them or store them in an air tight container.