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In a medium saucepan, combine the sugar with the water and simmer over moderate heat, without stirring, until an amber caramel forms, about 8 minutes. Add the shallots and stir well, then add the Cognac. Cover and simmer over low heat, stirring occasionally, until the caramel dissolves and the shallots are soft, about 10 minutes. Spread the mixture in a large dish and refrigerate for 20 minutes.
In a large skillet, melt the 4 tablespoons of butter. Add the chicken livers and season with salt and pepper. Cook over high heat until firm on both sides and deep pink in the center, about 3 minutes per side. Transfer the livers and their juices to a shallow dish and let cool to room temperature. Refrigerate the livers for 20 minutes.
In a food processor, combine the livers with the shallot mixture and process until pureed. With the machine on, gradually add the remaining 3 sticks of butter until thoroughly incorporated. Strain the puree through a fine sieve set over a large bowl. Season with salt and pepper.
Line a 4-cup terrine or loaf pan with sheets of plastic wrap, leaving a 3-inch overhang all around. Scrape the puree into the terrine and cover tightly with the overhanging plastic wrap. Refrigerate until very firm, at least overnight and for up to 5 days. Unwrap the terrine and turn it out on a platter. Serve chilled, with cornichons and sourdough toasts.