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1
Bring 3 cups water to a boil in a large pot. Add lobsters and cook until shells are red but meat is not yet cooked through, about 5 minutes. Remove lobsters; reserve cooking liquid. When cool enough to handle, crack lobsters over a large bowl to catch liquid; separate meat from shells.
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2
Cut meat into bite-size pieces; set aside meat and shells separately. Combine liquid from shells with cooking water in a large measuring cup; you should have 3 cups liquid (add water if needed). Set liquid aside. If lobsters are female, remove roe; set aside. (The roe sac is dark green when raw, red when cooked. The lobsters are half-cooked, so it will be green with pink around the edges.) Discard soft, pale green tomalley.
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3
Melt 2 tablespoons butter in a wide heavy pot over medium heat. Add shallot and cook, stirring occasionally, until shallot is soft, about 5 minutes. Add lobster shells and cook, turning occasionally, until lightly browned in spots and very fragrant, about 8 minutes. Remove pan from heat and add brandy. Return to heat and cook until brandy has almost evaporated. Add lobster cooking liquid and thyme sprigs and simmer until liquid is reduced by one-third, 8-10 minutes. Strain stock; discard solids in strainer. You should have about 2 cups stock.
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4
Melt remaining 2 tablespoons butter in same wide heavy pot over medium heat. Add roe, if using; mash and stir into butter until roe turns red, about 30 seconds. Add lobster meat; cook until just heated through, about 1 minute. Add stock and bring to a simmer. Remove stew from heat; stir in half-and-half. Season with salt, a small pinch of pepper, and a dash of Tabasco. Let cool; cover and chill overnight to allow flavors to meld.
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5
Rewarm stew gently over low heat, stirring occasionally (do not simmer or the lobster will become tough and cream may separate). Ladle into warm bowls and serve.
