Display Select Categories In Sidebar

On a few of my websites I'm trying to declutter my sidebar by removing the full list of categories and replacing it with something similar to:

makealivingwriting.com - On there, they have categories on the sidebar, but only key ones that really stand out.

I'd like to do something similar to that but I'm not sure if stock WordPress or Jetpack supports such a thing or if I need an additional plugin.

Thanks very much in advance!