Field-effect transistors have been widely used to study electronic transport and doping in colloidal quantum dot solids to great effect. However, the full power of these devices to elucidate the electronic structure of materials has yet to be harnessed. Here, we deploy nanodielectric field effect transistors to map the energy landscape within the bandgap of a colloidal quantum dot solid. We exploit the self-limiting nature of the potentiostatic anodization growth mode to produce the thinnest usable gate dielectric, subject to our voltage breakdown requirements defined by the Fermi-sweep range of interest. Lead sulfide colloidal quantum dots are applied as the active region, and are treated with varying solvents and ligands. In an analysis complementary to the mobility trends commonly extracted from field effect transistor studies, we focus instead on the subthreshold regime and map out the density of trap states in these nanocrystal films. The findings point to the importance of mapping comprehensively the bands and the gap structure within real quantum solids, and they suggest a new focus in investigating quantum dot solids with an aim towards improving optoelectronic device performance.