EVA's website UI/UX is horrible.

This is exactly what I was facing a while back. Though my situation was using EVA points to book an award ticket for UA. Here's the thing I can tell you. Whatever flights you can search for and the results returned under "Award Ticket-Select Itinerary" may not be actually bookable EVEN if you have enough miles.

Yup. You still have to call in to confirm. Give them a bunch of dates and they will search individually to tell you what's available.