I don't remember if the request for the sesame covered, bagel shaped rolls was ever given, but in case it wasn't, I found one in an Israeli cookbook: Yerushalayim shel mat'amim / [by] Rina Valero. Steimatzky, c1984. [From the style, I see the librarian in me is showing] The book is in Hebrew.

Place flour, margarine, oil, water, Arak, and spices into a bowl and mix until it is formed inot dough.

Take a walnut size piece of dough and roll into a rope about 1/3 inch in circumference and about 5 1/2 inches long, then join the two ends together forming a bagel. Continue until all the dough is used up.

Brush with egg white and sprinkle with sesame seeds.

Bake 45 minutes in a pre-heated moderate oven [350 degrees F]

Cool. Store in a closed container.

I've never made them but I would spray the cookie sheet with Pam or something similar to prevent sticking. From making real bagels, I would take the dough, poke a hole in the middle and stretch it into the proper shape and size. This way the ends won't come apart.

I ask our Israeli friends to let us know how large is the package of self-rising flour [kemah tofe'ah]

As to the "mahleb" I think it could be safely omitted if you can't get it.