Place racks in the upper and lower thirds of the oven and preheat the oven to 400 degrees F. Generously coat two baking sheets with cooking spray. Set aside.

Place the carrots, onions, and garlic in a large bowl. Drizzle with the olive oil, then sprinkle with 1 teaspoon salt, cumin, and pepper. Toss to evenly coat, then spread in a single layer on the prepared baking sheets, ensuring that the vegetables do not crowd one another. Roast for 25 to 30 minutes, turning twice throughout, until the vegetables are tender and browned. Let cool on the pans for 10 minutes.

While the vegetables cool, drain the juice from the tomato cans into a small bowl or measuring cup. Set aside.

Working in two batches, add half of the roasted vegetables to a food processor fitted with a steel blade or to a blender. Add 1 can of the drained tomatoes. Purée until smooth, then pour the purée into a 4-quart or larger heavy-bottomed pot, such as a Dutch oven. Repeat with the remaining vegetables and tomatoes, then add to the pot.

Stir in the reserved tomato juices, basil, yogurt, and remaining 1/2 teaspoon salt. Bring to a gentle simmer and let cook for about 10 minutes, until fully heated through. Taste and add additional salt and/or pepper as desired. Serve warm, topped with fresh basil and/or additional Greek yogurt.