  This paper introduces a new class of right-angled Coxeter groups with totally
disconnected Morse boundaries. We construct this class recursively by examining
how the Morse boundary of a right-angled Coxeter group changes if we glue a
graph to its defining graph. More generally, we present a method to construct
amalgamated free products of CAT(0) groups with totally disconnected Morse
boundaries that act geometrically on CAT(0) spaces that have a treelike block
decomposition.
