  A quantum phase transition (QPT) in a heavy-fermion (HF) compound may destroy
the Fermi liquid groundstate. However, the conditions for this breakdown have
remained obscure. We report the first direct investigation of heavy
quasiparticle formation and breakdown in the canonical system
CeCu$_{6-x}$Au$_x$ by ultraviolet photoemission spectroscopy at elevated
temperatures without the complications of lattice coherence. Surprisingly, the
single-ion Kondo energy scale $T_K$ exhibits an abrupt step near the quantum
critical Au concentration of $x_c=0.1$. We show theoretically that this step is
expected from a highly non-linear renormalization of the local spin coupling at
each Ce site, induced by spin fluctuations on neighboring sites. It provides a
general high-temperature indicator for HF quasiparticle breakdown at a QPT.
