Heat the oil in a Dutch oven over medium-high heat. Add the sausage and cook, stirring occasionally, until browned, 1 to 2 minutes. Transfer to a plate.
Add the carrots, onion, ¼ teaspoon salt, and ¼ teaspoon pepper to the pot. Cook, stirring, until the vegetables begin to soften, 4 to 5 minutes.
Add the tomato paste and cook, stirring, until it is slightly darkened, 1 to 2 minutes.
Add the broth, farro, and 2 cups water and bring to a boil. Reduce heat to medium and simmer until the farro is just tender, about 15 minutes.
Stir in the chickpeas, chard, sausage, ½ teaspoon salt, and ½ teaspoon pepper. Simmer until the chard is tender, 3 to 5 minutes.