  Quantum Hall edge channels can be combined with metallic regions to
fractionalize electrons and form correlated impurity models. We study a minimal
device, that has been experimentally achieved quite recently, with two floating
islands connected to three edge channels via quantum point contacts in the
integer quantum Hall regime. At high transparency of the quantum point
contacts, we establish a mapping to the boundary sine-Gordon model and thereby
reveal the nature of the quantum critical point. We deduce from this mapping
universal expressions for the conductance and noise, in agreement with the
experimental findings, and discuss the competition between Kondo-like screening
of each individual island and the cooperative transfer of electrons between
them. We further predict that the device operated at finite voltage bias
produces fractional charges $e^* =e/3$ and propose a generalization to $N$
islands with the fractional charge $e^* =e/(N+1)$.
