My all time favourite big band album has to be "Sinatra at the Sands", from way back in 1966. Recorded in front of a live audience at the Sands Hotel in Las Vegas, he was backed by the Basie Band, conducted and arranged by Quincy Jones.
For my money, this recording includes THE definitive versions of many Sinatra classics, including: Come Fly With Me, I've Got you Under My Skin, One for My Baby, You Make Me Feel So Young, Angel Eyes, It Was a Very Good Year and Fly Me to the Moon. Even the fantastic 1974 Madison Square Gardens concert with Woody Herman pales in comparison.
Frank was in great voice, and the band was studded with many of the legendary players associated with Basie, including Marshall Royal, Eddie "Lockjaw" Davis, Sonny Payne, Sweets Edison and the rock solid Freddie Green. In my opinion, the only downside to this recording is some of Frank's ad lib "asides" and brief monologues, which run the gamut from lame to cringingly dated. So fast forward through those parts.
Here's a link to the full hour-plus recording. Unfortunately, it's audio only, but it's still great:
I hope you enjoy it, and if you're not a Sinatra fan, you will be after listening to that!