In a large pot, combine the cardoons, lemon juice, and water to cover. Bring to a boil and cook until the cardoons are tender and easily pierced with the point of a paring knife, about 45 minutes. Drain. Allow to cool and slice into matchsticks. Set aside.
Bring 6 quarts of water to a boil and add 2 tablespoons of salt. Add the spaghetti and cook until tender yet al dente, about 10 minutes.
Meanwhile, in a 12 to 14 inch sauté pan, heat the olive oil until almost smoking. Add the garlic and toast until light golden brown. Add the cardoons and toss over high heat. Drain the pasta, reserving ½ cup of the pasta water and add the pasta to the sauté pan. Toss over high heat to coat the pasta with oil, adding some of the pasta water if necessary to “loosen” the sauce. Add the chives, season aggressively with salt and pepper and divide evenly among four warmed pasta bowls. Sprinkle castelmagno over each at the end and serve immediately.