Meanwhile, cook the eggplant. If possible, cook the eggplant on a grill/BBQ or sitting over a gas burner (you may want to cover around the burner with foil to catch any drips. Alternatively, preheat the oven to 430F/220C, prick the eggplant all over with a knife and place it on a baking sheet or in a baking dish. Roast for around 40 minutes. In all cases, you are looking for the inside of the eggplant to become very soft and if over grill/flame the skin should be well charred. Once cooked, leave the eggplant to cool down so that it is safe to handle when opened up.
Once cooled, cut open the eggplant skin and scoop out the flesh. Either roughly chop and mash the flesh for a coarser texture, or blend up for a smoother paste. In both cases, once roughly chopped, try to drain off as much liquid as you can as this can make it more sour. If you want a really smooth puree, you can press the mixture through a fine strainer to remove the seeds.
Melt the butter in a medium-small saucepan over a medium heat then once melted, add the flour. Stir as it cooks a couple minutes and becomes gently nutty smelling.
Gradually add the milk and stir or whisk in after each addition to make a smooth paste. Cook a minute or two more, as needed, to thicken slightly.
Remove from the heat then add the cheese and stir through so that it mixes in and melts. Add the eggplant puree and stir to mix through evenly. Season with a little salt and pepper to taste.
To serve, either spread some of the eggplant sauce over the bottom of individual plates or all of it on the bottom of a serving platter. In both cases, spread it out with a slight shallow area in the middle to hold the stew. Add the lamb stew in the middle of the eggplant sauce and serve with bread on the side.