The U.S. citizen marries the foreign fiance in the USA or in Thailand, the foreign fiance goes back to their country, the U.S. citizen applies to the Service Center for an I-130 spousal visa petition.
When the US citizen receives the notice of receipt from the service center for the I-130 petition, the US citizen files an I- 129F petition to the Chicago Center address for a K3 visa. Chicago Center sends this to the Missouri Center, and after Missouri Center approves the petition, it is forwarded to the foreign U.S. consulate.
The rest of the K3 spousal petition is similar to a K1 fiance visa process. The K3 spouse files for adjustment of status after entering the U.S., however has 2 years in which to do so. (The visa is a 2-year multiple re-entry visa)
The K3 visa is specifically intended for use by U.S. citizens who are sponsoring their foreign national spouse for immigration. It was created to address the long processing times for immigration visas at the USCIS. The K3 visa is therefore actually a "non-immigrant" visa, meaning that receipt of a K3 visa does not result in the spouse becoming a legal permanent resident immediately upon their arrival in the U.S. However, the K3 does allow the spouse to travel to the U.S. and live with their U.S. citizen husband or wife while they are waiting for their immigration application to sponsor their spouse to process through. They can also bring their unmarried children who are under 21 years old along using the related "K4" visa (however, please note that ordinarily it will be necessary for the U.S. citizen to file separate I-130 petitions for those minor children at some point in order for them to become legal permanent residents).