Weir use of “langsam”

In a sentence such as

  • “Ich frage mich langsam, wenn die Burger endlich serviert werden”

and

  • “Mir ist langsam langweilig”.

Does this use of langsam function like “beginning to” or “starting to”, as in “I’m starting to wonder when the burgers will be served”?

(Asked by Andrew)