Warm the clarified butter and milk in a small saucepan over low heat or in the microwave just until the butter is melted (not to exceed 115°F). Add the yeast and stir to dissolve.
In the bowl of a stand mixer fitted with the hook attachment, or by hand in a large bowl, combine the sugar, flour, mahleb, anise, arak, nutmeg, sesame seeds, and salt. Slowly add the butter mixture and mix on low speed or by hand until a cohesive dough forms. Increase the speed on the mixer to knead the dough for five minutes, or by hand on the counter for 10 minutes, to form a smooth, soft dough.
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 wax paper, parchment, or plastic wrap. Divide the dough into 18 pieces by cutting or squeezing off balls about 3 oz each (2 ½ inches wide; the size can be larger or smaller, to your liking). Place the balls on the prepared area about 2 inches apart, and cover with another towel. Let the balls rise for 30 minutes.
Heat the oven to 325°F, 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, tapping out any excess flour from the mold.
Lightly coat a dough ball with flour, and flatten the dough into a flat circle. Press the dough into the mold with some force with your fingers and palm numerous times without lifting the dough from the mold. Push in the dough 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. If using your hands to shape and design the top of thd 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. Transfer the shaped dough with the design side facing up on the prepared sheet pan.
Repeat this process with the remaining dough, arranging the molded dough about 2 inches apart with six at per pan. Bake one sheet at a time, re-using the pans and parchment once they are finished.
Bake for 25-30 minutes, until golden brown. Transfer the ka'ak to cooling racks until cool.
Make the glaze while ka’ak 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. Place parchment paper under cooling racks to catch glaze drips.
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 on cooling racks and allow 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.
Notes
Make Ahead
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.
Filling Kaak with dates
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.
Traditional Glaze
An alternative traditional glaze is to combine 1 tablespoon butter, 1/4 cup half and half, and 1/2 cup granulated sugar in a small saucepan over medium low heat. Bring just to a boil, then reduce the heat and simmer for 1 minute. Remove from the heat, add 1 teaspoon rose water and stir to combine. Dip the entire little loaf of warm kaak into the glaze when it is just out of the oven. Set on racks to cool and dry.