We observed two full orbital phase curves of the transiting brown dwarf KELT-1b, at 3.6 µm and 4.5 µm, using the Spitzer Space Telescope. Combined with previous eclipse data from Beatty et al. (2014), we strongly detect KELT-1b's phase variation as a single sinusoid in both bands, with an amplitude of 964 ± 43 ppm at 3.6 µm and 979 ± 46 ppm at 4.5 µm, and confirm the secondary eclipse depths measured by Beatty et al. (2014). We also measure noticeable Eastward hotspot offsets of 28.4±3.5 degrees at 3.6 µm and 18.6±4.8 degrees at 4.5 µm. Both the day-night temperature contrasts and the hotspot offsets we measure are in line with the trends seen in hot Jupiters (e.g. Crossfield 2015). This general agreement is despite the fact that KELT-1b should have an atmospheric radiative timescale substantially longer than in a typical hot Jupiter. We therefore suggest that nightside clouds are playing a noticeable role in modulating the thermal emission from these objects, based on: 1) the lack of a clear trend in phase offsets with equilibrium temperature, contrary to a recent suggestion of an offset trend by , 2) the sharp day-night transitions required to have non-negative intensity maps, which also resolves the inversion issues raised in Keating & Cowan (2017), 3) the fact that all the nightsides of these objects appear to be at roughly the same temperature of 1050 K, while the dayside temperatures increase linearly with equilibrium temperature, and 4) the trajectories of these objects on a Spitzer color-magnitude-diagram, which show colors only explainable via nightside clouds.