You will need a medium-sized bread bowl, two large baking sheets, and a pastry brush.
In a medium-sized bread bowl, dissolve the yeast and sugar in the warm water. Add 2 to 3 cups flour, a cup at a time, stirring constantly in the same direction to help activate the gluten, until a thick dough begins to form. Then stire 100 times in the same direction, about 1 minute. Let rest for 10 minutes.
Add the salt, butter, and mahleb, stir, and continue to add more flour, using a wooden spoon to stir it in. When the dough will no longer take any more flour, turn it out onto a lightly floured surface. Knead for 8 to 10 minutes, or until smooth and elastic, adding flour only when necessary. Rinse out the bread bowl, oil lightly, and return the dough to the bowl. Cover with plastic wrap and allow to rise until doubled in volume, approximately 1-1/2 hours.
Punch down the dough and divide into 16 pieces. On a lightly floured surface, roll each piece under your palms into a cigar-shaped rope 6 to 7 inches long. (You can also roll them in the air between your palms, letting the bread hang down vertically from your hands.) Pinch together the ends of each rope to form an oval-shaped loop. Place the ka#39;kat on lightly oiled baking sheets, at least 1/2 inch apart. Cover and let rise for 30 minutes.
Place the racks in the upper part of the oven, and preheat the oven to 400deg; F.
Just before baking, brush the breads with egg wash and sprinkle on the sesame seeds. Bake for 20 minutes, or until nicely browned; switch the baking sheets after 10 minutes. Cool slightly on a rack, then wrap in a cloth to keep warm. Serve warm.
Makes 16 oval bread rings about 3-1/2 inches across.