Maybe you already have the bakery information, or you no longer check these message boards, but anyway...
I went here:http://www.chicagoancestors.org/#tab-tools
then I clicked on Business A-D for the 1923 Chicago directory.
You didn't specify North or South Kedzie, but I think the Campbell's Soup factory was on the south side.
I see under Bakers-Retail
Dygdon John, 3815 S Kedzie av Also at the same address:
Kolodziej Joseph, 3815 S Kedzie av
That is probably the place, since there wasn't any other bakeries listed within 8 or 10 blocks of 3815 S. Kedzie.