I am only a first year graduate student, but I am very interested in mathematics education. My own approach to teaching is very much problem based: give students interesting problems which lead to the development of the concepts you want them to have. Even if they can't come up with all of the needed concepts on their own, if you give it to them after they have wrestled with a problem they will be much more likely to be able to apply the concept in novel situations in the future. Why couldn't this approach be carried through in a math grad situation? Design a sequence of problems, varying in difficulty, which in total cover need material from most of the "first year curriculum".
Before writing this off as a crazy idea, I would like to point out Cornell's vet school. They use exactly the model given above: Every week or two there is a new case. In each of your classes (anatomy, pharmacology, radiology, ...etc) you cover general information which is pertinent to the case of the week, but it is up to you and your team to do research, come up with a diagnosis and a method of treatment. So all of the classes you take are integrated together in the context of solving some real problems. Cornell is turning out some amazing vets. Why couldn't the same model work for mathematics?
