Heat 2 tablespoons of olive oil in a pot over medium-high heat until simmering.
Add eggplant, season with salt and pepper, and cook. Stir occasionally until browned in spots, about 2 minutes. Transfer to a large bowl.
Add 2 tablespoons of olive oil to a pot. Add zucchini, season with salt and pepper, and cook. Stir occasionally until browned in spots, about 2 minutes. Transfer to the bowl with eggplant.
Reduce the heat to medium. Add 1 tablespoon olive oil and onion, season with salt and pepper, and cook. Stir occasionally until softened and just beginning to brown, about 6-8 minutes.
Add garlic, thyme, and bay leaf. Cook until fragrant, about 30 seconds.
Add tomato and bell pepper. Gently stir to combine. Bring to a simmer, then turn down heat to medium-low. Simmer, stirring occasionally, for about 20 minutes. Shorter cooking time will leave vegetables in larger, more distinct pieces; longer cooking times will break vegetables down into a stew.
Add spinach, cook for 2-3 more minutes. Remove bay leaf and thyme sprigs. Just before serving, stir in the basil. Taste and season with salt and pepper as needed.
Serve, sprinkling with more basil and drizzling with olive oil.