Cook the eggplant - Transfer the eggplant pieces to a large bowl. Sprinkle with 1-2 teaspoons of salt and transfer to a large plate lined with paper towels. Microwave, uncovered, for 5-8 minutes, or until eggplant is dry to the touch and starting to shrivel. Set aside (see notes).
Cook the onions, potato, and eggplant - Heat a large Dutch oven or pot over medium heat. Add 2 tablespoons of olive oil and onions to the pot mixing well to combine. Cook the onions, stirring often, for 2-3 minutes. Add the potatoes and eggplant and cook, stirring frequently, for 4-6 minutes, or until onions are softened and the surface of the potatoes are translucent (they will not be fully cooked).
Work in the tomatoes and make a broth - Push the potatoes and onion to the side of the pot. Add the tomato paste and 1 tablespoon of olive oil to the other side. Cook the tomato paste, whisking continuously, until a brown fond (the browned bits) forms on the bottom of the pot (see notes). This will take approximately 2 minutes. Still stirring, add the water scraping the brown bits from the bottom of the pot. Add the canned tomatoes with their juice and fresh chopped tomatoes. Mix well. Bring to a boil.
Simmer - Reduce heat and bring to a simmer. Cook until the potatoes are tender and soft and the eggplant is broken down, approximately 25 minutes (the eggplant will help thicken the broth).
Cook the zucchini and bell pepper - As the stew simmers, add the remaining tablespoon of olive oil to a large skillet over high heat. Allow your skillet to get really hot before adding the zucchini, yellow squash, and bell pepper. Sprinkle with salt and cook, stirring occasionally, until just softening.
Add the herb pesto to the zucchini and bell pepper - Create a well in the middle of the zucchini and bell pepper. Add the herb pesto and cook until fragrant (oh my, goodness, it will smell so good!) before stirring to combine with the zucchini and bell peppers. Remove from heat.
Combine - Once potatoes are soft, add the zucchini and bell pepper mixture to the pot. Remove from heat and mix well to combine. Cover and allow flavors to combine for at least 10 minutes before serving. Serve with freshly toasted bread.