How to Make It
In a large pot of salted boiling water, cook the pasta until almost al dente. Drain, reserving 1 1/2 cups of the pasta cooking water.
Meanwhile, in a large skillet, combine the olive oil, anchovies and the 3 halved Calabrian chiles. Cook over moderately low heat, stirring, until the anchovies dissolve, about 5 minutes. Stir in the garlic and radishes and cook for 1 minute. Discard the chiles.
Add the pasta and 1 cup of the reserved pasta cooking water to the skillet, along with the 2 crumbled chiles, 2 tablespoons of the chives, 2 teaspoons of the lemon zest and the lemon juice. Season with salt and pepper and cook, tossing occasionally, until the pasta is al dente and most of the liquid is absorbed; add more of the pasta cooking water if necessary to cook it through. Stir in the 1/2 cup of grated cheese and transfer to serving bowls. Garnish with the remaining 1 tablespoon of chives and 1 teaspoon of lemon zest and serve, passing additional grated cheese at the table.