Motivated by the work of Knutson, Lam, Speyer and Snider, Bruhat atlases, introduced by He, Knutson and Lu are a way of locally modeling a stratification of a manifold on the stratification of Schubert cells by opposite Schubert varieties. He, Knutson and Lu described Bruhat atlases on many interesting varieties, including partial flag varieties and on wonderful compactifications of groups. We will discuss some results toward a classification of varieties with Bruhat atlases, focusing on the 2-dimensional toric case. In this case, the answer may be stated in terms of the moment polygon of the toric surface, which one should first slice up, then put toppings on, much like one would do while preparing a pizza. Speaker(s): Balazs Elek (Cornell)