1. Preheat oven to 350 degrees. Line 2 large cookie sheets with unbleached parchment paper.
2. In the bowl of an electric mixer fitted with the paddle attachment, beat together the butter and sugar on medium-high speed until light and fluffy, about 3 minutes. With the motor running on low speed, add 3 of the eggs to the butter mixture, 1 at a time, allowing each to be incorporated before adding to the next. Then add the vanilla and mix until just combined.
3. In another large mixing bowl, whisk together the flour, baking powder, baking soda, baker’s ammonia, salt and cinnamon. With the motor of the electric mixer running on low, slowly add the dry mixture to the egg mixture one-third at a time, pausing after each addition. Mix just until the dough no longer sticks to the sides of the bowl, being careful not to over-mix, or your cookies will be tough.
4. Very lightly oil a marble baking slab or a large plastic cutting board as well as your hands. Using a tablespoon as a measure, scoop out a piece of the dough. Roll the dough between your hands and/or on the oiled surface until it resembles a long Tootsie Roll, about 8 ½ inches long. Fold in half and twist together 2 to 3 times to form a cookie twist. Press the ends together and lay on the prepared cookie sheet. Repeat with the remaining dough. Keep some paper towels nearby in case your hands or the surface bet too greasy.
5. In a small mixing bowl, lightly beat the remaining egg. Using a pastry brush, brush the egg over the top of each cookie. Evenly sprinkle the cookies with the sesame seeds (if using), pressing them very lightly into the dough. Transfer to the oven to bake for 25 to 30 minutes or until golden brown. Remove the cookies, cool for 3 to 4 minutes on the cookie sheets and transfer to cooling racks to cool completely. Koulourakia will keep in an airtight container at room temperature for up to 1 week.