Wash and dice the eggplant into roughly 1-inch cubes. Quarter and thinly slice the zucchini and yellow squash. Halve the cherry tomatoes, dice the onion, and mince the garlic cloves.
Heat half of the olive oil (2 tablespoons) in a large skillet or Dutch oven over medium heat. Add the diced eggplant and sprinkle with 1/2 teaspoon kosher salt. Cook, stirring occasionally, until the eggplant begins to soften and lightly brown, about 6-8 minutes. Remove the eggplant and set aside.
Add the remaining olive oil to the skillet. Toss in the diced onion and sauté until translucent and fragrant, about 4 minutes. Add the minced garlic and cook for another 30 seconds, stirring continuously to prevent burning.
Stir in the zucchini and yellow squash slices. Cook for 3-4 minutes until they start to soften but still hold their shape.
Return the sautéed eggplant to the skillet. Add halved cherry tomatoes and drained garbanzo beans. Sprinkle Italian seasoning and paprika evenly over the mixture.
Pour in the low-sodium vegetable broth and gently stir everything together. Bring to a simmer, then reduce heat to low. Cover the skillet and cook for 15-20 minutes, stirring occasionally, until the vegetables are tender and the flavors meld.
Taste and add more salt if needed. Remove from heat and sprinkle torn fresh basil over the stew for a vibrant finish.