Ever since Outlook.com (formerly Hotmail) has offered IMAP, it has had an annoying bug when used with the Mac Mail application. Two copies of each message are stored in the Sent folder. The Outlook.com program manager acknowledged that this was due to Outlook.com’s SMTP server saving a copy of a message. He wrote that he’s exploring the option to disable this feature, but this was over one year ago. It doesn’t look like this issue is going to be addressed any time soon by Microsoft.
The simplest solution to avoid the duplicate sent message is to uncheck “Store sent messages on the server” in your Outlook.com preferences.
The drawback to this solution is that mails sent from other devices (such as an iPhone or the Outlook.com web site) will show not show up in the Sent mail folder (with the airplane icon). The Sent mail folder is a local (non-IMAP) folder. The IMAP Sent mail folder is still there, at the bottom of the mailbox list.