The long rise, both for the first and second proofs, is essential. To do the first rise overnight in the refrigerator, pull back to 1 ½ tablespoons yeast, cover the dough as usual, and chill in the refrigerator for about 8 hours. Proceed with making the balls, and let them rise in a warm spot until doubled, about an hour. To fill the ka'ik with date paste, see the method here. Note the molded imprint is not quite as defined when the cookie is filled.
Proof the yeast by dissolving it in 1/4 cup of warm water with a tablespoon of the sugar. After about 10 minutes, the yeast will activate, becoming creamy and foamy.
Warm the clarified butter and milk in a small saucepan over low heat or in the microwave just until the butter is melted.
In the bowl of a stand mixer fitted with the hook attachment, or by hand in a large bowl, combine the yeast mixture with the remaining cup of sugar, flour, mahleb, anise, arak, nutmeg, sesame seeds, and salt. Slowly add the butter and milk and mix on low speed or by hand until dough forms. Increase the speed on the mixer to knead the dough for eight minutes, or by hand on the counter for 10 minutes.
Lightly oil a large bowl with the olive oil. Coat the dough in the bowl and cover with plastic wrap, then a clean kitchen towel. Set the dough in a warm spot to rise until at least doubled in size, which can take 2 to 3 hours.
To create a warm setting for the balls to rise again, place a kitchen or bread towel on the counter or on sheet pans and cover with plastic wrap. Divide the dough into 18 pieces by cutting or squeezing off balls about 2 ½ inches wide (the size can be larger or smaller, to your liking). Place the balls on this about 2 inches apart, cover with more plastic wrap and another towel. Let the balls rise for 30 minutes, uncovering the dough for the last 15 minutes.
Heat the oven to 325 degrees, with a rack in the center of the oven. Line two sheet pans with parchment paper and if you have more sheet pans, double the pans to prevent overbaking the bottom of the ka'ik. Dust the counter and molds lightly with flour.
Take a dough ball and lightly coat it with flour, flattening the dough into a flat circle. Press the dough into the mold with some force numerous times without lifting the dough from the mold, and pushing the dough up that falls over the edges as you go.
To unmold the dough, use gravity as your friend and turn the mold over. Gently peel the dough away from the edges and it will fall out into your hand.
Repeat this process with the remaining dough, baking six at a time per pan, then re-using the pans and parchment once they are finished. If using your hands to shape the dough, flatten each ball with the palm of your hand. Pinch the edges five or six times around the circle and poke with the tines of a fork over the top. Place on the prepared sheet pans.
Bake the ka'ik for 25-30 minutes, until golden brown. Transfer the ka'ik to cooling racks until cool. Place parchment paper under the cooling racks to catch the glaze drips.
Make the glaze while ka’ik bakes. Combine the confectioners' sugar with rose water, water, and corn syrup, stirring until the glaze falls in a ribbon off of the spoon. Add drops of water if needed to get the correct consistency. Cover with plastic wrap until you're ready to use the glaze, stirring again to smooth it out.
Dip the tops of the ka'ik in the glaze, letting the excess glaze run off of the ka'ik back into the bowl. Place the ka'ik back on the racks and let the glaze harden for at least one hour.
Serve immediately, or store in airtight containers for up to 5 days. The cookies will not be as soft over time, which is perfect for dipping them in coffee. To soften them, warm them in the microwave for 15-20 seconds right before serving.