Hello, Small world. I was plotting a similar road trip yesterday. In several weeks we plan to approach the GrossglocknerStrasse from the NW, coming down from Chiemsee -- so the routing will be a bit different. Like you, we will then head back north, by Salzburg, to Mondsee. Disclaimer: this is all based on websites and maps; I expect more reliable advice will follow from the Austria Forum regulars.
I would suggest using GoogleMap or ViaMichelin to investigate something like the following:
1. Route from Salzburg to Zell Am See, then the short drive down to the northern end of the GS at Ferleiten. Rts 21 to 78 to 311 looks promising. Others with real experience will hopefully give you better suggestions.
2. Travel the GS to southern end at Heiligenblut.
3. (The following is our scheme) from there: Rts 107 and 106 east to Seeboden; get on A10 for a relatively straight shot back to Salzburg.
You should have plenty of time with a moderately early start. Looks like the drive to the start will be about 2 hours. Others on the Forum have suggested 2 hours+ for the GS drive, but dining and admiring the views probably make this highly variable. The return to Salzburg will be 2.5 hrs or so, depending on route. (One is a partial backtrack on the GS.) More info on the Grossglocknerstrasser website: https://www.grossglockner.at/gg/en/index