Argh! I've been living here for 11 years now, and cannot tell you the pain not having a Dunkin Donuts has caused me.
To be honest, there is no good doughnut shop in town. We have two Dixie Creme Donut shops, but....I've never been impressed. Same for the grocery store offerings.
As for other breakfast pastry options, your best bet is going to be Uprise Bakery. http://www.uprisebakery.com/ They have good pastries and bread, as well as fantastic soup, sandwiches, and decent pizza. They're in the same buiding as the independent video/movie theater, and I've heard tell you can bring in Uprise Food and Beer into the theater.
Columbians will try and sell you on The Upper Crust bakery....but don't give in. The product is bought frozen and proofed and baked on site. If that's your thing....well...I think Panera does it better.
The Upper Crust
907 Elm St, Columbia, MO 65201
6710 Clayton Rd, Saint Louis, MO 63117