Combine 1 quart of cold water and 2 tablespoons of salt in a large bowl. Add the diced eggplant and let soak for 30 minutes. Drain well in a collander.
In a large saute pan, heat 1/3 cup of the olive oil over medium-high heat until very hot, and add the drained eggplant. If your saute pan is not big enough to hold all the eggplant in one layer, do this step in more than one batch. Cook the eggplant until well browned and tender, 8 to 10 minutes. Stir often during the last half of cooking, and reduce the heat as needed to keep the eggplant from burning. Transfer to a dish and set aside.
In a small saucepan, bring the chicken stock to a simmer.
Combine 2 tablespoons of oil, the shallots, garlic and anchovies in a 3 quart, heavy-bottomed saucepan or skillet. Place over medium heat and stir to cook, without coloring, about 2 minutes. Add the rice, and 1 teaspoon of salt, and stir with a wooden spoon until the rice is coated with the oil. Add the wine and bring to a boil over medium high heat; cook the rice, stirring constantly, until it absorbs the wine.
Ladle 1/2 cup of the simmering broth into the saucepan and stir until absorbed. Continue with the rest of the broth, adding a 1/2 cup at a time and letting each addition become absorbed completely into the rice before adding more liquid. The constant stirring allows the rice to release its starch into the cooking liquid, resulting in the characteristic risotto creaminess. Count on approximately 20 to 25 minutes for the rice to be cooked al dente.
Fold in the eggplant and cook until heated through, about 30 seconds. Add the mint, swirl in the butter and half the cheese,and season with a 1/2 teaspoon of salt and pepper. Spoon the risotto onto a warm platter or individual bowls, sprinkle with the remaining pecorino, and serve immediately.