I think I am understanding your question, you want your Toggl projects to match what is listed as a task in Asana. I work in the marketing department of my organization and we have 6 departments that we do work for.
There is no auto-fill per say. You would have to start/stop down your task list to get all the entries for the day. I usually have a browser with both Asana and Toggl open and jump back and forth.
The way I have the Asana/Toggl integration set-up, my Asana tasks become the description in Toggl. I use the projects in Toggl to categorize work by department. I rarely use the Chrome extension but, I do use the Asana extension frequently, which should auto-populate the description in Toggl.
Occasionally there is a delay w/ my internet or something is connecting and I will copy and paste my entries from Asana directly into Toggl so the match.
I hope that helps.