Heat the oil in a Dutch oven over medium-high heat. Add the chorizo and cook, stirring occasionally, until browned, 2 to 3 minutes. Transfer to a plate.
Add the carrots, onion, 1/4 teaspoon salt, and 1/4 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 tender, 20 to 25 minutes. Stir in the chickpeas, chard, chorizo, 1/2 teaspoon salt, and 1/2 teaspoon pepper. Simmer until the chard is tender, 3 to 5 minutes.