That's interesting. It indeed seems unnatural because I've never heard anybody say it like that - my impression is that you generally can't just use both particles with the same subject for the same sentence (or at least not like this.) But maybe that's just because of my inexperience. Do you have a real-life example of this pattern being used?

(It's also strange because I typoed 牛肉 in my earlier comment, and milk is difficult to eat. Sorry for the confusion.)

Almost - you would write it 私は「私が」牛乳を食べます, and obviously 私は doesn't serve a grammatical purpose, it only has an indicative function.

I'm amused by the transition from beef to milk.

Yes, thanks for the correction tor3!