I can do this with fastmail, though fastmail is a subscription (like $5/month? IIRC, mine auto renews every 2 years so not sure). I have my primary email setup as <firstname>@<lastname>.org. If you set your dns records correctly with them, that allows you to use without any ahead of time setup <randomtag>@<firstname>.<lastname>.org. Setting a different tag where I have <firstname> is can be done too, but you need to set those up individually.

replying to emails, I can change <randomtag> to whatever I want.

They also offer random domains that you can setup burners under, though that does involve some ahead of time setup.