This is a pretty broad category. You need to narrow down your parameters. Moreover, a composition of MY choice would not be a composition of your choice. What decade interests you? What do you mean by "The Jazz Era?" Have you been given any specific details? Are you referring to jazz in America? You might look first at some of Ken Burns's wonderful series on jazz. Both DVDs and a book are available. Moreover, you might want to listen to some different styles of jazz and some different composers.
One who comes to mind immediately is Duke Ellington. His orchestra was famous as the house band at the Cotton Club. He composed not only jazz but sacred music as well. This would seem to be a contradiction in terms, but he was equally adept at both and is remembered for both. He was also known for writing compositions for various members of his band that would feature them as soloists.
You might need to do a little more research and re-post your question.