  We report a detailed study of the microscopic effects of Cu intercalation on
the charge density wave (CDW) in 1\textit{T}-Cu$_x$TiSe$_2$. Scanning tunneling
microscopy and spectroscopy (STM/STS) reveal a unique, Cu driven spatial
texturing of the charge ordered phase, with the appearance of energy dependent
CDW patches and sharp $\pi$-phase shift domain walls ($\pi$DWs). The energy and
doping dependencies of the patchwork are directly linked to the inhomogeneous
potential landscape due to the Cu intercalants. They imply a CDW gap with
unusual features, including a large amplitude, the opening below the Fermi
level and a shift to higher binding energy with electron doping. Unlike the
patchwork, the $\pi$DWs occur independently of the intercalated Cu
distribution. They remain atomically sharp throughout the investigated phase
diagram and occur both in superconducting and non-superconducting specimen.
These results provide unique atomic-scale insight on the CDW ground state,
questioning the existence of incommensurate CDW domain walls and contributing
to understand its formation mechanism and interplay with superconductivity.
