Self-assembled nanocrystal solids show promise as a versatile platform for novel optoelectronic materials. Superlattices composed of a single layer of lead-chalcogenide and cadmium-chalcogenide nanocrystals with epitaxial connections between the nanocrystals, present outstanding questions to the community regarding their predicted band structure and electronic transport properties. However, the as-prepared materials are intrinsic semiconductors; to occupy the bands in a controlled way, chemical doping or external gating is required. Here, we show that square superlattices of PbSe nanocrystals can be incorporated as a nanocrystal monolayer in a transistor setup with an electrolyte gate. The electron (and hole) density can be controlled by the gate potential, up to 8 electrons per nanocrystal site. The electron mobility at room temperature is 18 cm²/(V s). Our work forms a first step in the investigation of the band structure and electronic transport properties of two-dimensional nanocrystal superlattices with controlled geometry, chemical composition, and carrier density.