  We investigate the controllability of an infinite-dimensional quantum system:
a quantum particle confined on a Thick Quantum Graph, a generalisation of
Quantum Graphs whose edges are allowed to be manifolds of arbitrary dimension
with quasi-$\delta$ boundary conditions. This is a particular class of
self-adjoint boundary conditions compatible with the graph structure. We prove
that global approximate controllability can be achieved using two physically
distinct protocols: either using the boundary conditions as controls, or using
time-dependent magnetic fields. Both cases have time-dependent domains for the
Hamiltonians.
