Start by piercing the eggplant with a fork about 10 times and pop it into a 400 degree oven for about 30-35 minutes until wilted and cooked through. Flip it over half way through the cooking process. Leave it to cool.
Peel and dice the potato, onion and carrot and put in a large saucepan covered with water and bring to a boil.
Cook for about 10 minutes or until all the veggies are soft. Reserve about 1/2 a cup of the cooking liquid and then strain the veggies.
Add in all the cheese sauce ingredients into a high powered blender along with the cooked potato, carrot and onion and blend. Add in some of the reserved cooking liquid- about 2 Tbsp at a time until it just comes together. You want the mixture to be thick and cheese-like.
Add your cheese sauce back into the pot you cooked the veggies in and whisk away! The tapioca starch will help the cheese to thicken into a nice dip.
In the meantime, split your eggplant down the middle and scoop out the inside flesh, discard the skin. Add the eggplant into the cheese mixture and continue to whisk. The whisking will break up the eggplant into small pieces.
Continue whisking until it is the consistency you want. Serve it up in a nice bowl and it is ready for dipping!
If you want you can bake the dip after you've cooked it on the stove to form a crust, but it is not necessary.