To make the risotto: In a medium-sized saucepan, heat the stock until almost a boil. Turn the heat to very low and stir in the saffron.
In a 10 to 12-inch saucepan, melt 3 tablespoons of the butter. Immediately stir in the rice and cook over medium-high heat. Using a wooden spoon, stir continuously until the rice is coated in the butter and begins to turn translucent, about 2 minutes. Be careful not to let the rice burn. Keeping the heat at medium-high, begin ladling the hot broth into the rice, about 2 ladles at a time. Constantly stir with the wooden spoon until the rice has almost absorbed all of the broth. Continue adding the broth bit by bit until the rice is al dente or tender, about 18 to 20 minutes. Once the rice is cooked, season with salt and pepper. Remove from the heat.
Preheat oven to 350 degrees.
To make the stuffed eggplants: Par-bake eggplants in the oven for 25 minutes. Cool. Leave 1-inch on tops and bottoms of eggplant and cut lengthwise down the middle, making little pockets.
Combine the rice, garlic, mint, basil, cheese and egg yolk together in a medium bowl. Mix until well combined. Stuff an equal amount of the stuffing into each little pocket. Be sure to overstuff the eggplants. Use toothpicks to hold together, if necessary.
In a 12-inch saute pan, heat the oil over medium-high heat until hot but not smoking. Fry the eggplant, in batches, until the skins begin to wrinkle.
In the meantime, heat the tomato sauce in another 12-inch frying pan. Add the sugar and stir to combine. Season, to taste, with salt and pepper. Stir in 1 cup of water.
Transfer the cooked eggplants to the pan with the tomato sauce, and spoon the sauce over the eggplants. Simmer over medium heat for 10 minutes. Turn the eggplants over and cook for 5 more minutes. Serve hot or at room temperature with extra sauce and basil.
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.