Preheat oven to 350 degrees F.
If you're using raw sesame seeds, place the seeds in a single layer on a baking sheet. Place in the oven for about 3 minutes, watching carefully. They should just begin to brown lightly. Set aside to cool completely.
Lower the oven temperature to 300 degrees F. Line cookie sheets with parchment paper.
Using an electric mixer, cream the sugar, butter and eggs until very light, about 5 minutes. Sift the flour, baking powder, and salt. Add to the butter mixture and stir with the spatula until combined. Add the vanilla. Stir in the cooled sesame seeds. Drop the batter by 1/2 teaspoonfuls onto the prepared cookie sheets, 1 inch apart. It takes just a dab of batter to produce quarter-sized cookies. They will spread into perfect circles during baking.
Bake until very brown but not burned at the edges, about 14 to 15 minutes. Important: Let the cookies cool completely on the parchment paper, then peel them away from the paper. Store between layers of waxed paper in an airtight container. These cookies freeze well in tins. They will crumble in a plastic freezer bag.