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My Fair Lady, recorded by Shelly Manne & His Friends and released in 1956 by Contemporary Records, is the first album ever made consisting entirely of jazz versions of tunes from a single Broadway musical. It was an instant hit and became one of the best-selling jazz albums of its day (Wikipedia, https://goo.gl/i568Na).

AllMusic Review by Scott Yanow:
This trio set by Shelly Manne & His Friends (consisting of the drummer/leader, pianist André Previn, and bassist Leroy Vinnegar) was a surprise best-seller and is now considered a classic. Previn (who is really the main voice) leads the group through eight themes from the famous play, including “Get Me to the Church on Time,” “I’ve Grown Accustomed to Her Face,” “I Could Have Danced All Night,” and “On the Street Where You Live.” The result is a very appealing set that is easily recommended (https://goo.gl/SEVNce).

Track listing:
All tunes by Frederick Loewe. (Lyrics of the original songs were by Alan Jay Lerner.)

“Get Me to the Church on Time” – 4:11
“On the Street Where You Live” – 5:37
“I’ve Grown Accustomed to Her Face” – 3:21
“Wouldn’t It Be Loverly” – 5:31
“Ascot Gavotte” – 4:17
“Show Me” – 3:40
“With a Little Bit of Luck” – 6:01
“I Could Have Danced All Night” – 3:00

Personnel:
Shelly Manne – drums, leader
André Previn – piano
Leroy Vinnegar – bass
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Released: 1956
Recorded: August 17, 1956
Contemporary Records Studio, Los Angeles, California
Length: 35:56
Label Contemporary Records
C 3527
Producer Lester Koenig