An Outlook user had a problem: their account was migrated to the new Outlook.com server and they wanted to change the display name. Their account is setup in Outlook as an EAS account but the issue also affects Exchange users: there is no display name field.
When you use Outlook desktop with an Outlook.com account, or use Outlook on the web, there is only one way to change the display name used on your email. You'll need to log into your Microsoft account and edit your profile.
You can log directly into the Microsoft account, click Your info then Edit name
If you are logged into Outlook.com, you can get to this screen from Options, Connected Account, Manage or choose primary address, or by selecting View account under your user photo, then select Edit Name.
If you have multiple aliases, each one will use this display name.
Remember: if your account was moved to the new server you should remove the account from Outlook and use auto account setup to add it back as a Microsoft Exchange account. It's a better experience and supports all Outlook features.