Heat 2 tablespoons of the olive oil in a large sauti pan over medium-high heat until hot. Add the mushrooms and cook, without moving them, until brown on one side, about 1 minute. Continue to sauti for another 1 or 2 minutes. Lower the heat to medium, add the leek, season with salt and pepper, and sauti until the leek is soft but not brown, about 2 minutes. Add 1 tablespoon garlic and sauti for another minute. Add 1 teaspoon of the thyme and stir. Scrape the vegetables into a large bowl. Give the pan a quick rinse and scrub if anything has stuck and burned.

In the same pan, heat another 1 tablespoon of the olive oil over medium-high heat until hot. Add the bell pepper and sauti for about 1 minute. Add the yellow and green zucchini and cook until they release their water and turn translucent but have not colored, about 2 minutes. Add the remaining 2 teaspoons thyme. Season with salt and pepper. Spread the vegetables in the bowl with the mushrooms so they cool quickly and retain their color.

Heat the remaining 1/2 tablespoon olive oil in the same sauti pan over medium-high heat until hot. Add the remaining 1 teaspoon garlic. Sauti briefly until lightly colored, then add the tomatoes. Season with salt and pepper. Bring to a boil and simmer until the mixture thickens, about 5 minutes. Add the spinach and toss until it wilts into the tomatoes. Scrape into the bowl with the other vegetables and stir so the mixture cools a bit. Add the basil and grate the parmesan over the mixture. Toss until well mixed. Serve warm or at room temperature.