The Aztec diamond, under the uniform measure on domino tilings, is one of the classic examples of an exactly solvable model in probability and statistical mechanics. Its rich geometric features—such as limit shapes and arctic boundaries—have long made it a cornerstone of integrable probability. More recently, variants of this model with doubly periodic weights have revealed that much of the underlying structure persists far beyond the uniform case and can be used to uncover new behaviors—such as regions with smooth disorder—that were previously out of reach. At the heart of these models lies a birational map that encodes their integrable character. In this talk, I will present a new disordered version of the Aztec diamond obtained by placing Gamma-distributed weights on its edges. Remarkably, these random weights preserve just enough structure to keep the model integrable, This allows us to provide rigorous confirmation of predictions from the physics literature—such as the absence of a phase transition in the free energy—and explain the emergence of n^2/3 fluctuations near the turning points. Remarkably, these turning points are related to certain integrable polymer models. The presentation is aimed at a broad mathematical audience and is based on on joint work with Roger van Peski.