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We use FIRE simulations to study disk formation in z~0, Milky Way-mass galaxies, and conclude that a key ingredient for the formation of thin stellar disks is the ability for accreting gas to develop an aligned angular momentum distribution via internal cancellation *prior* to joining the galaxy. Among galaxies with a high fraction (>70%) of their young stars in a thin disk (h/R~0.1) we find that: (i) hot, virial-temperature gas dominates the inflowing gas mass on halo scales (>~20 kpc), with radiative losses offset by compression heating; (ii) this hot accretion proceeds until angular momentum support slows inward motion, at which point the gas cools to T~10^4 K or less; (iii) prior to cooling, the accreting gas develops an angular momentum distribution that is aligned with the galaxy disk, and while cooling transitions from a quasi-spherical spatial configuration to a more flattened, disk-like configuration. We show that the existence of this "rotating cooling flow" accretion mode is strongly correlated with the fraction of stars forming in a thin disk among a sample of 17 z~0 galaxies spanning a halo mass range of 10^10.5 solar masses to 10^12 solar masses, or a stellar mass range 10^8 solar masses to 10^11 solar masses. Notably, galaxies with a thick disk or irregular morphology do not undergo significant angular momentum alignment of gas prior to accretion and show no correspondence between halo gas cooling and flattening. Our results suggest that rotating cooling flows (or, more generally, rotating subsonic flows) that become coherent and angular momentum-supported prior to accretion onto the galaxy are likely a necessary condition for the formation of thin, star-forming disk galaxies in a LambdaCDM universe., 17 pages, 11 figures. Comments welcome. Accompanying interactive visualization at zhafen.github.io/rotating-cooling-flows