Melody Gardot - Which albums to buy?

I would go with "The Absence " but since you like live recordings and the "Live in Europe" is really good, that would be my choice for you.

"Live" has a lot of the live gymnastics that you will love or hate. Check out youtube and see what you think.

The recording quality is fine on the live recording. I just don’t care for the style of the live compositions and her performance. Completely subjective and just my personal preferences at play. 

I find it interesting to follow the 'arc of development' of an artist. Gardot is easy to follow in this regard as she hasn't been shy about taking chances and her catalog isn't wide (yet) . Her next album should be illustrative.

Nice songs. I own ’One and Only Thrill.’ Shame about the quality of the vinyl. However I can ignore some crackles if the recording is good.
After playing Melody’s album, I played a 70s recording by Roger Whittaker. It was etched and crystalline with the finest highs and an open clear transparency. It is a joy to hear the most minute details- the color of percussive elements, tiny bells, the clarity of his baritone etc. The Gardot recording is dull, muted and veiled by comparison. No doubt digital. If only artists knew what they are being cheated out of.

@noromance   I read your post above just before receiving "My One and Only Thrill" CD today.  I note that your comments refer to the LP of that release.  Specifically, I bought the European release which includes a second CD of "Live in Paris".  

The sound quality of the main CD is superb in every way.  Melody's voice is right in the room!  The orchestra and other supporting instruments were well placed and clear and clean.

I'm looking forward to listening to the live tracks later today.

It's a shame the sound of the LP is so poor.  It's also surprising.  Perhaps the European release was engineered and mastered more expertly.  Both CDs are mastered by Bernie Grundman.