While the meat is cooking, roast the eggplants.
You can do this either on the stove on a non stick pan or in the oven. Whichever method you choose to use, make sure it is really hot before you put the eggplants.
Before starting to roast the eggplants cut off the stalks and the green leaves of each eggplant. Make several holes with a thin knife or a toothpick on every side of the eggplants.
Make little cuts close to the part where they used to have stalks, to make it easier to peel after they are roasted.
Cook until they are well done. You can understand whether they are ready by slightly pushing down, they will look like burnt and start looking like empty.
Then peel the eggplants. They will be very hot obviously, so watch out. There are two alternative methods to peel: Cut the eggplants vertically in the middle. Scoop out the white part with at small spoon or hold them from the once stalky part. Peel them downwards using the small cuts you had done in the first place.
Cut the eggplant contents into tiny parts.
In a separate pot or skillet melt the butter.
Add the flour and cook mixing quickly and continuously with a whisk until you can start smelling the flour.
Add the chopped eggplants, cook mixing for another 1-2 minutes.
Add the water, milk and salt, cook until it combines well and gets a thick texture.
Grate the cheese with the thinner side of the grater. Close the heat of the eggplant mixture and add the cheese. Mix well again.