  We present a method for measuring the masses of galaxy clusters using the
imprint of their gravitational lensing signal on the cosmic microwave
background (CMB) temperature anisotropies. The method first reconstructs the
projected gravitational potential with a quadratic estimator and then applies a
matched filter to extract cluster mass. The approach is well-suited for
statistical analyses that bin clusters according to other mass proxies. We find
that current experiments, such as Planck, the South Pole Telescope and the
Atacama Cosmology Telescope, can practically implement such a statistical
methodology, and that future experiments will reach sensitivities sufficient
for individual measurements of massive systems. As illustration, we use
simulations of Planck observations to demonstrate that it is possible to
constrain the mass scale of a set of 62 massive clusters with prior information
from X-ray observations, similar to the published Planck ESZ-XMM sample. We
examine the effect of the thermal (tSZ) and kinetic (kSZ) Sunyaev-Zeldovich
(SZ) signals, finding that the impact of the kSZ remains small in this context.
The stronger tSZ signal, however, must be actively removed from the CMB maps by
component separation techniques prior to reconstruction of the gravitational
potential. Our study of two such methods highlights the importance of broad
frequency coverage for this purpose. A companion paper presents application to
the Planck data on the ESZ-XMM sample.
