  Quantum systems can be used as probes in the context of metrology for
enhanced parameter estimation. In particular, the delicacy of critical systems
to perturbations can make them ideal sensors. Arguably the simplest realistic
probe system is a spin-1/2 impurity, which can be manipulated and measured
in-situ when embedded in a fermionic environment. Although entanglement between
a single impurity probe and its environment produces nontrivial many-body
effects, criticality cannot be leveraged for sensing. Here we introduce instead
the two-impurity Kondo (2IK) model as a novel paradigm for critical quantum
metrology, and examine the multiparameter estimation scenario at finite
temperature. We explore the full metrological phase diagram numerically and
obtain exact analytic results near criticality. Enhanced sensitivity to the
inter-impurity coupling driving a second-order phase transition is evidenced by
diverging quantum Fisher information (QFI) and quantum signal-to-noise ratio
(QSNR). However, with uncertainty in both coupling strength and temperature,
the multiparameter QFI matrix becomes singular -- even though the parameters to
be estimated are independent -- resulting in vanishing QSNRs. We demonstrate
that by applying a known control field, the singularity can be removed and
measurement sensitivity restored. For general systems, we show that the
degradation in the QSNR due to uncertainties in another parameter is controlled
by the degree of correlation between the unknown parameters.
