How to Make It
Combine oil and ham in a large Dutch oven, and cook over medium, stirring often, until ham fat is rendered, about 6 minutes. Remove ham with a slotted spoon, and set aside.
Increase heat to medium-high. Add carrots, onion, leek, and garlic to Dutch oven; cook, stirring occasionally, until vegetables are tender but not browned, 6 to 8 minutes. Add chopped tomatoes, chopped mint, salt, cumin, and black pepper. Cook, stirring occasionally, until tomatoes have released their liquid and are completely broken down, 45 to 50 minutes.
Working in batches, transfer soup to a blender. Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over opening. Process until very smooth, about 1 minute. Pour mixture through a fine wire-mesh strainer into a bowl; discard solids. Serve hot, sprinkled with reserved ham and mint leaves, drizzled with oil, and topped with fig jam or fresh figs.