For all who are first trying to set this up and not seeing the icons in Asana: it was not clear to me that the Toggl integration steps were performed in the settings opened from the cog wheel inside of the browser extension window – not at https://track.toggl.com. Your address bar when adding this integration should look like either “moz-extension://…” (for FireFox) or “chrome-extension://…” (for Chrome).
Also note this also works for current Ubuntu 20.04 versions of those browsers.