***Chicago is a great song. I once wondered if it and My Kind of Town were written around the same time.***
No. "Chicago" was written in 1922 and became a hit of the "swinging 20s". Sinatra's recording however was made just a few weeks before the album sessions in 1957.
"My Kind Of Town" was written for Sinatra (or rather, the movie where he introduced it in, Robin & The Seven Hoods) in 1963.
It should be noted that "On The Road To Mandalay" did not appear on the original album release in the United Kingdom, actually being replaced by "Chicago".
The Mandalay song is based on a poem by the famous British poet and author Rudyard Kipling, and Kipling's heirs thought the recording by Sinatra did not keep up with the high standards of Kipling's written legacy.
So through a lawsuit they prevented Capitol to issue the recording anywhere within what was at the time left of the British Empire, while they did not succeed in banning the song from being released on the US issues of the album.
Sinatra reacted cleverly, instantly adding the song to his concert repertoire (the ban did not include any live performances), and as can be heard on the Capitol/Blue Note release of his 1959 Australian performances for instance, he never missed commenting on the affair with ironic remarks about Kipling's estate and their policy when introducing the song on stage.
A few years later, the case was settled and "Mandalay" got its well-deserved British release as well. Strange affair, isn't it... how could any note sung by Frank Sinatra harm any legacy, in this case?