Heat a big glug of olive oil in a large pot. Sauté the onion until softened, then add the garlic and cook for another minute. Stir in the rice and sauté for a couple of minutes until lightly toasted.
Pour in half of the vegetable stock along with the tomatoes and paste. Simmer, stirring frequently, until the rice absorbs the liquid.
Gradually add the remaining stock, continuing to simmer until the rice is cooked through, about 20 minutes. Stir in the vegan parmesan or nutritional yeast, parsley and season with pinches of salt to taste. Remove from the heat and allow to cool.
Slice the eggplants lengthwise into long pieces. Shallow fry in the sunflower oil on each side until lightly browned. Drain well on absorbent paper.
Grease a 23-24 cm diameter ring cake tin. Line the tine with the fried eggplant slices, overlapping them slightly. Spoon about half of the cooked rice over the eggplant, pressing down evenly.
Distribute the mozzarella over the rice layer, then top with the remaining rice, smoothing it down to reach the top of the tin.
Bake at 190°C for about 30 minutes or until the rice forms a slight crust on top. Cool a little before carefully inverting the timballo onto a serving plate. You may need to tap on the tin to help release it. Serve warm.