There is nothing like thyroxine specialist. You can always go to word general physician. This is a very common condition and can easily be dealt with by medications. When hypothyroidism medication usually started with eltroxin 100 mg and will be continued.  slowly slowly doses will be reduced but this medication will have to be continued for lifetime. In many cases at has been seen that often after taking the medication for 2 to 3 years hypothyroidism gets controlled​ and then there is no need to take medication but you need to continue for at least 3 years.