As the name implies, the threads can be done in parallel if you have enough cooks and kitchen space.
Bread the eggplant slices in flour, egg, and panko. Fry until tender - not more than a minute or two per side.
Cook the rice.
When the potatoes and carrots are cooked, add the roux to the pot with veggies, and bring to a boil briefly to thicken.
Serve in layers - rice on the bottom, eggplant in the middle, curry on top.
Noah and I made this on 2015-03-01. We considered baking the eggplant after frying it, but ended up deciding not to, and I think it worked out well. The eggplant ended up firm but not rubbery, and had an interestingly vegetal taste.
A silicone spatula works well for stirring the roux.
Made this 2016-02-28, accompanied by miso soup with tofu to add some protein. Noteworthy: if you use whole wheat flour for the roux, the color change is much harder to observe. The wheat germ quickly turns dark red, but that doesn't mean the roux is done. You need to look at the color of the froth. I recommend using white flour for this reason alone; whole wheat flour didn't affect the taste.