The UT permit used to do just that. Unless your home state only accepts resident permits from another state if it has reciprocity with UT their permit is valid. I carry with a UT non res permit in NC myself.
But, last year UT changed their permit laws so that now you have to have your own state's permit before you can get their non res permit. Nice people as those Mormons are, that will however NOT invalidate existing permits (which are valid for a generous 5 years) but you can't renew them unless you first get the local permit. Which stinks for me because NC requires citizenship for a CCW permit and I am only a resident permanent legal alien and not rich enough to sue them to make it otherwise (many states have dropped similar restrictions after being sued).
Not exactly true. Only time you are required to get your home state permit before applying for the UT permit is if your home state honors the UT permit. This allows residents of states like CA, HI, NJ, IL, NY, etc. the ability to apply for the non-resident UT CFP and not submit a copy of their home state permit. Kudos to the state of UT for writing a law to do exactly what was intended to do, yet limited so it would not exclude otherwise legal law abiding citizens in the process.
A little history behind the change in the law. States like NM were concerned because their residents were acquiring and carrying under a UT CFP in the state of NM. The NM permit is substantially more costly than the UT CFP. This resulted in a loss of revenue for the state of NM, which also prompted them to stop recognizing the UT CFP. To protect their relationships with other states, and the further erosion of states from their impressive list of recognizing states, the law was drafted and passed by UT lawmakers.