  Reconstructed maps of the lensing convergence of the cosmic microwave
background (CMB) will play a major role in precision cosmology in coming years.
CMB lensing maps will enable calibration of the masses of high-redshift galaxy
clusters and will yield precise measurements of the growth of cosmic structure
through cross-correlations with galaxy surveys. During the next decade, CMB
lensing reconstruction will rely heavily on temperature data, rather than
polarization, thus necessitating a detailed understanding of biases due to
extragalactic foregrounds. In the near term, the most significant bias among
these is that due to the thermal Sunyaev-Zel'dovich (tSZ) effect. Moreover,
high-resolution observations will be available at only a few frequencies,
making full foreground cleaning challenging. In this paper, we demonstrate a
solution to the foreground bias problem that involves cleaning only the
large-scale gradients of the CMB temperature map. We show that the data
necessary for tSZ-bias-free CMB lensing maps already exist in the form of
large-scale measurements of the CMB across multiple frequencies by the Planck
and WMAP satellite experiments. Specifically, we show that the bias to halo
masses inferred from CMB lensing is eliminated by the utilization of clean
gradients obtained from multi-frequency component separation involving Planck
and WMAP data, and that special lensing maps for galaxy cross-correlations can
be prepared with only a small penalty in signal-to-noise while requiring no
masking, in-painting, modeling, or simulation effort for the tSZ bias. While we
focus on cross-correlations, we also show that gradient cleaning can mitigate
biases to the CMB lensing autospectrum that arise from the presence of
foregrounds in temperature and polarization with minimal loss of
signal-to-noise.
