In addition to St. Anna, a whipped cream frosted cake from any of these would fit your needs:
Oak Mill - European Bakery
2204 W North Ave, Chicago - (773) 252-4400 (see website for additional locations)
Bombon Cafe - Mexican Bakery
38 S Ashland Ave, Chicago - (312) 733-8717 (see website for additional locations, including Ogilvie)
Lutz Continental Cafe & Pastry Shop - European Bakery
2458 W Montrose Ave Chicago, IL 60618-1648 - (773) 478-7785