  Ly$\alpha$ nebulae ubiquitously found around z>2 quasars can supply unique
constraints on the properties of the Circumgalactic Medium, such as its density
distribution, provided the quasar halo mass is known. We present a new method
to constrain quasar halo masses based on the line-of-sight velocity dispersion
maps of Ly$\alpha$ nebulae. By using MUSE-like mock observations obtained from
cosmological hydrodynamic simulations under the assumption of maximal quasar
fluorescence, we show that the velocity dispersion radial profiles of
Ly$\alpha$-emitting gas are strongly determined by gravity and that they are
thus self-similar with respect to halo mass when rescaled by the virial radius.
Through simple analytical arguments and by exploiting the kinematics of
HeII1640\.A emission for a set of observed nebulae, we show that Ly$\alpha$
radiative transfer effects plausibly do not change the shape of the velocity
dispersion profiles but only their normalisation without breaking their
self-similarity. Taking advantage of these results, we define the variable
$\eta^{140-200}_{40-100}$ as the ratio of the median velocity dispersion in two
specifically selected annuli and derive an analytical relation between
$\eta^{140-200}_{40-100}$ and the halo mass which can be directly applied to
observations. We apply our method to 37 observed quasar Ly$\alpha$ nebulae at
3<z<4.7 and find that their associated quasars are typically hosted by
~$10^{12.16 \pm 0.14}$ M$_{\odot}$ haloes independent of redshift within the
explored range. This measurement, which is completely independent of clustering
methods, is consistent with the lowest mass estimates based on quasar
auto-correlation clustering at z~3 and with quasar-galaxies cross-correlation
results.
