When I'm looking for an award flight, I search always research my flights as one ways. I'll only use the round trip or advance tool when I'm ready to book, since all award flights are priced as one-ways.
1. I search based off of origin and destination, and a date near when I want to go.
2. When I get to the results, I click on Flexible Calendar, which is just a 30-day calendar,.
3. Then one week at a time, I'll click on a departure date, and select my filters and sort orders. Being EWR based, I always try and connect through LAX or SFO if possible for the lie-flats.
The rates in the calendar are the lowest rates for all flights on that day, You can't filter down the calender; You'll have to set your filters and sort order manually for each day.
I'm a 100% leisure traveler and plan trips 6+ months out. While our day of the week for departing might be solid, we always have our choice of which week we go.