Is it possible to do back-to-back broadening assignments?

Is is 'possible'? Absolutely. Is it 'likely'? Depends.

This is a conversation you'll have to have with your assignment manager*. It's all going to depend on what shortages are occurring at your grade. Suggest you give them a call and discuss with them.
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* https://www.hrc.army.mil/content/Enlisted%20Infantry%20Branch%20POCs

While it is true ( as others have stated) however it is not likely that a Branch Manager would approve it. The word broadening is only broadening as long as the service emeber returns to their career field and applies such skill sets as developed during the time away.

If we are talking career here I can unequivocally say with a few accepting, selection to the next grade by a board will look more favorable at a successful broadening tour and return home than they will for someone gone for six plus years even if successful. Having said that, if you excell at both and are rated above your peers and set sone sort of example as to be rated appropriatley then there is the acception.

So, only you know your capabilities.

Possible? Sure. Depends on what your Branch says. You may run the risk of missing "time on the line" which could bite your six. Talk with Branch, get their input and go from there. IMHO, do recruiting, then go back to the line for a couple of years, then go DS.