Heat 1 teaspoon oil over medium heat in a Dutch oven or soup pot. Add onion and cook, stirring often, until softened and lightly browned, 2 to 3 minutes. Add cabbage and minced garlic; cook, stirring often, until the cabbage has wilted, 2 to 3 minutes. Add broth, mashed beans and whole beans; bring to a simmer. Reduce heat to medium-low, partially cover and simmer until the cabbage is tender, 10 to 12 minutes. Season with pepper.
Shortly before the soup is ready, toast bread lightly and rub with the cut side of the garlic clove (lightly or heavily depending on taste). Divide toast among 8 soup plates. Ladle soup over the toast and sprinkle with cheese. Drizzle about 1 teaspoon oil over each serving. Serve immediately.