  Many-body correlations and macroscopic quantum behaviors are fascinating
condensed matter problems. A powerful test-bed for the many-body concepts and
methods is the Kondo model which entails the coupling of a quantum impurity to
a continuum of states. It is central in highly correlated systems and can be
explored with tunable nanostructures. Although Kondo physics is usually
associated with the hybridization of itinerant electrons with microscopic
magnetic moments, theory predicts that it can arise whenever degenerate quantum
states are coupled to a continuum. Here we demonstrate the previously elusive
`charge' Kondo effect in a hybrid metal-semiconductor implementation of a
single-electron transistor, with a quantum pseudospin-1/2 constituted by two
degenerate macroscopic charge states of a metallic island. In contrast to other
Kondo nanostructures, each conduction channel connecting the island to an
electrode constitutes a distinct and fully tunable Kondo channel, thereby
providing an unprecedented access to the two-channel Kondo effect and a clear
path to multi-channel Kondo physics. Using a weakly coupled probe, we reveal
the renormalization flow, as temperature is reduced, of two Kondo channels
competing to screen the charge pseudospin. This provides a direct view of how
the predicted quantum phase transition develops across the symmetric quantum
critical point. Detuning the pseudospin away from degeneracy, we demonstrate,
on a fully characterized device, quantitative agreement with the predictions
for the finite-temperature crossover from quantum criticality.
