We prove that the class of topological knot types that are both Legendrian
simple and satisfy the uniform thickness property (UTP) is closed under
cabling. An immediate application is that all iterated cabling knot types that
begin with negative torus knots are Legendrian simple. We also examine, for
arbitrary numbers of iterations, iterated cablings that begin with positive
torus knots, and establish the Legendrian simplicity of large classes of these
knot types, many of which also satisfy the UTP.