How to create a firebase rule to allow creating a child node with the same name as $uid as soon as user signs up

Question

def sign_up(email, password):
    # create user node with UID under NGO
    user = auth.create_user_with_email_and_password(email, password)
    uid = user['localId']
    db.child("NGO").child(uid).set({
        'email': email
    })
    return user
if __name__ == "__main__":
    config = {
    # API config
    }
    firebase = pyrebase.initialize_app(config)
    db = firebase.database()
    auth = firebase.auth()
    user = sign_up(email, password)

Now to make this code work, the only solution I have come up with so far is

{
  "rules": {
    "NGO": {
      ".read": false,
      ".write": true,
      "$uid": {
        ".read": "auth.uid === $uid",
        ".write": "auth.uid === $uid"
      }
    }
  }
}

This approach unfortunately gives write access to the entire NGO node to any $uid

What did not work (it can't create a new node named $uid itself but can read/write if it already exists) -

{
  "rules": {
    "NGO": {
      "$uid": {
        ".read": "auth.uid === $uid",
        ".write": "auth.uid === $uid"
      }
    }
  }
}

The RTDB structure looks like this