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Puree 1 cup chickpeas and 1/2 cup broth in a blender until smooth. Set aside.
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Heat olive oil in a large soup pot and sauté onion, carrot and celery for 6-8 minutes until vegetables are softened by not browned. Stir in garlic and cook for 1 additional minute. Add the remaining 2 cups of chickpeas, remaining 3 cups of broth, tomatoes (and their juices), tomato sauce, rosemary, salt and pepper. Stir to incorporate and bring to a boil. Reduce heat to medium-low and continue to cook for 15 minutes.
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Add farro. Increase heat and bring mixture to a boil. Once it boils, reduce heat and simmer soup (covered) until the farro is tender. This will take about 20 minutes.
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Discard the rosemary sprig if you used fresh rosemary. Add the chickpea puree and stir in the spinach. Cook for 1-2 minutes until spinach is wilted but still bright green.
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Serve with a sprinkling of grated Asiago cheese.