In a large Dutch oven over medium-high, heat 3 tablespoons of the oil until shimmering. Add the onion and garlic, then cook, stirring occasionally, until the onion begins to turn translucent, about 3 minutes. Add the tomato paste and cook, stirring constantly, until the the onion is well coated and the tomato paste is fragrant, 1 to 2 minutes. Add the carrots, potatoes, 2½ cups water, ½ tablespoon salt and ¾ teaspoon pepper, then bring to a simmer. Cover, reduce to medium and cook until a knife inserted into a potato meets no resistance, 15 to 18 minutes.