It isn't a common grammar, but since the rest of the sentence is perfectly fine we have to assume it is intentional and use context to glean the meaning. In this paragraph the author contrasts those who lead "privileged" lives to those who don't, so we can paraphrase the unusual part as if the person was keeping a daily log:
23:40 - It's now hour 11 of my (possibly 12-hour) shift, manning this deep-fat fryer.
I don't know how common it is for kitchen workers to do 11+-hour shifts, but that's really beside the point. The main focus is on the contrasting images of "sun-dappled yoga" and "interminable hours spent deep-frying food"
[Edit] To be clear, the phrase means that it's been ten (not eleven, thank you MathieuK) hours since the employee started the shift, and now they're on "Hour 11". It's not a measure of duration, nor does it imply how much longer they have to go before they finish.
Also, as Paul points out, "The 11th hour" is an idiomatic expression meaning "The latest possible time before it is too late (to make any change)". But that's probably best asked as a separate question since it's only indirectly related to this context.