Rinse the eggplants, pat dry and remove the stems. Cut each eggplant in half lengthwise and carve a diamond pattern into the flesh.
In a large frying pan, heat 2 tablespoons olive oil over medium heat. Place 4 eggplant halves with the cut side down and saute about 10 minutes. Flip eggplant halves twice. Then cook the other halves with 2 tablespoons olive oil in the same manner.
Brush a sufficiently large baking dish for all the eggplant halves with oil. Peel the shallots and cut into thin rings. Rinse the tomatoes, cut in half, and cut the flesh into cubes.
Remove the eggplant flesh with a spoon.
Heat 2 tablespoons of olive oil in a pan and saute onions, tomatoes and eggplant flesh for about 5 minutes. Simmer over medium heat and season with salt, sugar and pepper. Stir in the thyme and half of the parsley.
Spoon the mixture into the eggplant shells, leaving space for 1 egg in the middle of each. Crack an egg into each eggplant.
Sprinkle eggplant with remaining parsley and bake in a preheated oven at 200°C (approximately 400°F) for 15 minutes.