First, declare a custom_dir in mkdocs.yml if you haven't already:
theme:
custom_dir: docs/.overrides # Relative to mkdocs.yml. Can be whatever you want.
Then download Font Awesome, extract it, and move the contents of the svgs directory (not including the folder itself) to {CUSTOM_DIR}/.icons/fontawesome. For example, if your custom_dir is docs/.overrides, you would move the contents of the svgs directory to docs/.overrides/.icons/fontawesome. Everything else that came with the download is safe to delete.
Your custom_dir should look like this:
.
└── .icons/
└── fontawesome/
├── brands
├── duotone
├── light
├── regular
├── sharp-regular
├── sharp-solid
├── solid
└── thin
You can then use Font Awesome Pro icons in the same way you already do the free ones (e.g. :fontawesome-thin-user-secret: will work as written).