In a medium bowl, cover the chicken livers with the milk and let stand for 1 hour. Drain and pat completely dry.
In a large skillet, heat the oil until shimmering. Add the chicken livers and cook over high heat until lightly browned on the bottoms, about 2 minutes. Turn the livers. Add the shallot, garlic and thyme to the skillet and cook until the shallot and garlic are fragrant and softened, about 2 minutes. Add the sherry and cook, scraping up any bits stuck to the bottom of the pan, until the liquid is evaporated and the livers are cooked through, about 2 minutes. Scrape the mixture into a food processor and pulse until chopped. With the machine on, add the butter and puree until smooth. Scrape the chicken-liver butter into a bowl and season with salt and pepper.
In a large pot of generously salted boiling water, cook the pasta until just al dente. Drain the pasta, reserving 3/4 cup of the cooking water. Return the pasta to the pot. Add the chicken-liver butter, chives, tomatoes and the reserved pasta cooking water. Bring to a simmer over moderately low heat, tossing until the pasta is coated with the creamy sauce. Transfer to bowls and serve right away.
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