In a colander set in the sink, toss eggplant cubes with Kosher salt. Drain for 1 hour. Wipe eggplant dry with paper towels. While eggplant is draining, arrange green and red peppers on a baking tray. Broil 3" from heat until skins are evenly blistered red and charred about 15 to 20 minutes, turning occasiona;;y.
Seal hot peppers in a paper bag for 10 minutes. Peel, seed and cut peppers in strips, holding them over a bowl to catch any juices. In a 6 quart nonreactive pot, heat olive oil over medium heat. Add onions and cook until transparent, about 3-5 minutes, stirring often. Add eggplant and garlic. Cook until eggplant starts to brown, about 5 minutes, stirring often.
Add tomatoes, zucchini, peppers with any juices, potatoes and chicken broth combined with the pesto. Season with salt and pepper to taste. Mix gently. Bring to a simmer. Reduce heat. Cover and simmer over medium-low heat for 40 minutes until potatoes are tender. Combine Parmesean Cheese, chopped basil, and chopped parsley. Sprinkle atop stew.