Last I checked Minnesota recognizes the Utah permit. The Utah permit is available through the mail and there are several places in Iowa that offer the Utah training.
I don't recall the process exactly but I believe it starts with downloading the application and filling it out. I believe it must be notarized but the people offering the training tend to also be notaries. Fingerprints will need to be submitted. In my case the trainer supplied the fingerprint cards and took my fingerprints. The trainer will have to sign the application.
Minnesota is a shall issue state and the process sounds similar to the one from Utah. I don't have a permit from Minnesota but did some research as I have considered applying for one in the past. I ended up getting a Utah permit as that was the easier route.
The process for a Minnesota permit seems very similar to the Utah permit. The application can be downloaded from the internet, application must be signed by the trainer, and training must be from a state recognized trainer. The only real difference I can see in the process between the Utah and Minnesota permit process is that the application for the Minnesota permit must be submitted in person to a sheriff.
Since the application must be submitted in person it is rare to see training for the Minnesota permit offered in Iowa. There is a company based in Minnesota that does come to certain northern Iowa locations to offer training so if you live close to the border you might be able to find a class that is close. Utah permit classes have become real popular around here since it is valid for the Utah permit as well as the Florida and Iowa permits. The Florida permit is not valid in Minnesota but might come in handy for other states, such as Kansas. The Texas permit is recognized in Minnesota and many other states, is shall issue to non-residents, but the training must be obtained in Texas.
I hope this helps.