Create shared JS utility for refreshing To-Do counter

~"Category:Alert Management" has a nice method for manually syncing the To-Do counter: https://gitlab.com/gitlab-org/gitlab/blob/bba77591beca7fc738000b60e6bda24d6231ebe0/app/assets/javascripts/alert_management/components/sidebar/sidebar_todo.vue

The code for this will be reused in Category:Design Management (in a follow-up MR to !40222 (merged)).

This code seems reusable enough that it would be worth abstracting it into a shared utility for all to use!