Any doctor (or nurse, for that matter) can administer a mini-mental status test. It's quite brief. It involves things like "spell WORLD backwards" count up by adding/subtracting 7s, be able to repeat a list of objects named to you 10 minutes ago, etc. Your doctor might opt to do this before referring you to a specialist.

A doctor won't mind you raising this issue. But a doctor might hesitate to refer you, if he/she finds that you seem basically mentally competent.

A higher level of testing that does not take more than an hour is done by speech therapists. They do a lot of testing in nursing homes.

The gold standard would be done by a clinical psychologist. That's a lengthy, expensive series of tests. Your insurance would need a good reason to pay for that - like you having difficulties keeping yourself safe.