We present optimal measurements of the angular power spectrum of the XDQSOz catalogue of photometric quasars from the Sloan Digital Sky Survey. These measurements rely on a quadratic maximum likelihood estimator that simultaneously measures the auto- and cross-power spectra of four redshift samples, and provides minimum-variance, unbiased estimates even at the largest angular scales. Since photometric quasars are known to be strongly affected by systematics such as spatially-varying depth and stellar contamination, we introduce a new framework of extended mode projection to robustly mitigate the impact of systematics on the power spectrum measurements. This technique involves constructing template maps of potential systematics, decorrelating them on the sky, and projecting out modes which are significantly correlated with the data. Our method is able to simultaneously process several thousands of non-linearly correlated systematics, and mode projection is performed in a blind fashion. Using our final power spectrum measurements, we find a good agreement with theoretical predictions, and no evidence for further contamination by systematics. Extended mode projection not only obviates the need for aggressive sky and quality cuts, but also provides control over the level of systematics in the measurements, enabling the search for small signals of new physics while avoiding confirmation bias.