Actually UA does not allow stopovers. However, it allows an “Excursionist Perk”, which is a free one-way award within select multi-city itineraries. According to the UA MilagePlus rules, The origin and destination of the Excursionist Perk is within a single MileagePlus defined region. This is the UA example, which can be done for a total of 60K miles:
- ORD - LHR, 30K miles
- LHR - CDG, 0 miles
- CDG - ORD, 30K miles
UA considers Laos to be in South Asia, which includes Bangladesh, Bhutan, Brunei, Cambodia, Hong Kong, Indonesia, Laos, Macau, Malaysia, Myanmar (Burma), Philippines, Singapore, Thailand, and Vietnam. So the OP can’t do ORD-TPE, TPE-BKK-VTE, VTE-BKK-NRT-ORD, unless he/she is willing to do them as 3 individual 1-way awards.