Slice the carrots very thin, put them in a medium pot and add the water and 1/2 teaspoon salt. Bring to a boil, drop the heat so the water boils gently, and cover with a lid. Cook until the carrots are tender enough to mash with a fork, about 25 minutes.
While the carrots cook, bring a big pot of generously salted water to a boil and cook the pasta. Reserve about a quarter cup of the pasta cooking water and then drain the pasta. Put the cooked pasta back into the pot and keep it warm near the stove.
When the carrots are tender, put them along with any remaining carrot cooking liquid into a food processor fitted with a metal blade. Add the milk and puree until silky smooth. You can also do this with a blender.
Stir the carrot puree into the pasta. Set aside 3/4 cup of the cheese and add the remainder to the pasta along with the chives, remaining salt, and several grinds of black pepper. Taste and add more salt and pepper as needed. If the sauce seems too thick, stir in a tablespoon or two of the pasta water.