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.
The chicken liver butter can be refrigerated overnight. Bring to room temperature before using.