Experiments demonstrating the controlled growth of oxide heterostructures have raised the prospect of realizing topologically nontrivial states of correlated electrons in low dimensions. Here, we study heterostructures consisting of {111} bilayers of double perovskites separated by inert band insulators. In bulk, these double perovskites have well-defined local moments interacting with itinerant electrons leading to high temperature ferromagnetism. Incorporating spin-orbit coupling in the two-dimensional honeycomb geometry of a {111} bilayer, we find a rich phase diagram with tunable ferromagnetic order, topological Chern bands, and a C=±2 Chern insulator regime. Our results are of broad relevance to oxide materials such as Sr_{2}FeMoO_{6}, Ba_{2}FeReO_{6}, and Sr_{2}CrWO_{6}.