Although crucial to our understanding of nuclear structure, probes of gluonic components in the nucleus can be elusive, as gluons are accessed only indirectly in deep inelastic scattering. In 1989, Jaffe and Manohar identified a gluonic transveristy structure function $\Delta(x,Q^2)$ which is sensitive to exotic gluonic states in the nucleus, and is accessible via an inclusive measurement on a transversely polarized nucleus of spin greater than or equal to 1. We are developing an experiment to utilize Jefferson Lab's 12 GeV electron beam with a transversely polarized $^{14}$N target to mount a first measurement of the $\Delta$ structure function. The Jefferson Lab experiment will probe $\Delta$ from from $x$ of 0.3 to 0.05, but the vast kinematic reach of an electron-ion collider would allow a thorough probe of this quantity. We will discuss the impact of exciting new lattice QCD results on this quantity, our proposal to measure $\Delta(x,Q^2)$ at JLab, and what a measurement might look like at an EIC.