Transit in Dubai with prescribed methadone

While you haven't indicated whether you'll be going through immigration controls during your connection in Dubai, even in transit you would want to have a medical prescription with you, not just the medication. The medication has to be in its original packaging and in a quantity considered appropriate to personal prescribed use.

The UAE has strict laws (and severe penalties) regarding controlled substances, that can affect those in transit, not just those entering the country.

Gulf News January 2018 coverage of the Ministerial decree which lists banned and controlled medicines and narcotic drugs:

Watch out for these if you are bringing medicines to UAE

Travellers to the UAE are advised to consult the detailed list of the banned medicines and drug schedule and seek permission to carry these from the Registration and Drug Control Department of the ministry prior to commencement of the journey.

In case of controlled medicines for personal use, both travellers to the UAE and those transiting through the country have to carry a valid medical prescription from the country of origin. In case the original prescription is not available, travellers are advised to carry attested prescription. For those terminating their journey in the UAE, an additional authenticated certificate or permit from the relevant health authority is also required. In case of controlled medicines for personal use, the amount of medicines cannot exceed more than 30 days of treatment.

And Thailand has similar regulations, as it lists methadone among the controlled substances for which those under treatment are required to obtain a permit prior to entry.

Guidance for Travellers to Thailand under Treatment Carrying Personal Medications Containing Narcotic Drugs/Psychotropic Substances into/out of Thailand

Travellers to Thailand under treatment of these medications are required to obtain a permit (Form IC-2) issued by the Food and Drug Administration before travelling to Thailand.

The quantity of the medication transported into Thailand must not exceed 30 days of prescribed usage. To apply for the permit, please download the application form (Form IC-1) and supplement and submit the completed form along with copies of the medical prescription and the required certificate at least two weeks before the arrival date by:

Fax: (66) 2591 8471 or
Email: tnarcotics@fda.moph.go.th or
Mail:
Narcotics Control Division
Food and Drug Administration
Ministry of Public Health
Nonthaburi 11000, THAILAND

Methadone is considered a controlled narcotic in the UAE, and the relevant rules say:

Transit travelers have to comply with the required procedures when entering and leaving the UAE

"Prior approval from the Registration and Drug Control Department of the Ministry of Health" is required, even if you're just transiting. There is a security check for transit passengers, and while they're mainly looking for weapons and other threats to aviation security if you are just transiting and not going through immigration and customs, I wouldn't want to risk prison in Dubai.

The process to obtain that approval can be found on the Ministry's website.

As Giorgio notes, you'll need to comply with the relevant requirements for Thailand too.