  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. An effective two-band model of Zeeman-split j=3/2 states captures this
nontrivial band topology. Our results are of broad relevance to oxide materials
such as Sr2FeMoO6, Ba2FeReO6, and Sr2CrWO6.
