Place the oven tray or pizza stone on the lower rack of your oven. If you a using an oven tray, cover it with parchment paper. Preheat oven to 400F.
For the bell pepper spread, put the red bell pepper face down on the oven tray and brush with olive oil. Place the tray on the upper rack and broil for at least 15 minutes until bell peppers look somewhat burnt.
Put the bell peppers in a bowl and cover with plastic film until they have cooled down. Remove the peel with a knife. Using a mixer, puree the flesh with the rest of the olive oil. Season with salt and pepper.
In a large pan, heat the olive oil on medium-high heat and cook to eggplant slices until slightly golden. Season with salt and pepper. Set aside.
Roll out your dough into 4 circles and spread red bell pepper spread on them. Place the eggplant and cheese slices on top. Cook for about 6-8 minutes and take out off the oven.
Carefully break an egg at the center of each pizza. Cook for another 8 minutes, until the egg white is cooked but the egg yolk is still runny. Season with pepper and serve warm.