Hey there barneyhairball,
Thanks for reaching out to the Apple Support Communities. To get started, it would only be possible to select an existing unique, random email address if you have previously sent an email to that contact using the Hide My Email feature.
Check out the resources below for more information
Use Hide My Email in Mail on iPhone - Apple Support
When you subscribe to iCloud+, Hide My Email lets you send and receive email messages that forward to your real email account, to keep your real email address private. With iOS 15.2 or later, you can generate unique email addresses on demand in the Mail app 
.
Send an email with Hide My Email
1. Tap 
.
2. Add a recipient and subject for your email.
Note: You can only send a message using Hide My Email to one recipient at a time.
3. Tap the From field, tap it again, then tap Hide My Email. A new, unique email address appears in the From field.
Tip: When the recipient replies to an email you sent with Hide My Email, their reply forwards to your real email address. To set a Forward To address, see Create and manage Hide My Email addresses in Settings.
Use Hide My Email on Mac - Apple Support
Send an email using a unique, random address
When you write a new email message, Hide My Email allows you to create a unique, random address directly in the Mail app.
1. In the Mail app 
on your Mac, click the New Message button 
to compose a new message.
2. Add a recipient and subject for your message.
Messages you send from a unique, random address can only be delivered to a single recipient.
3. Move the pointer over the From field.
4. Click the pop-up menu that appears, then choose Hide My Email.
- If you’ve never sent this recipient an email from a unique, random address: A new address appears in the From field.
- If you previously sent this recipient an email from a unique, random address: The address used previously with this recipient appears in the From field.
5. Write your message, then when you’re ready, click the Send button 
.
Let us know if that's able to help.
Regards.
