  We propose a new method for probing inflationary models of primordial black
hole (PBH) production, using only CMB physics at relatively large scales. In
these scenarios, the primordial power spectrum profile for curvature
perturbations is characterized by a pronounced dip, followed by a rapid growth
towards small scales, leading to a peak responsible for PBH formation. We focus
on scales around the dip that are well separated from the peak to analytically
compute expressions for the curvature power spectrum and bispectrum. The size
of the squeezed bispectrum is enhanced at the position of the dip, and it
acquires a characteristic scale dependence that can be probed by
cross-correlating CMB $\mu$-distortions and temperature fluctuations. We
quantitatively study the properties of such cross-correlations and how they
depend on the underlying model, discussing how they can be tested by the next
generation of CMB $\mu$-distortion experiments. This method allows one to
experimentally probe inflationary PBH scenarios using well-understood CMB
physics, without considering non-linearities associated with PBH formation and
evolution.
