Place shallots, chicken livers, thyme, and 1⁄3 cup water in a frying pan on medium-high heat. Bring to a simmer and turn down to medium low. Simmer until chicken livers are cooked through, 3–5 minutes. Pour into a food processor. Add the salt, butter, cinnamon, and rum. Process until smooth. Transfer the pâté to your favorite small jars/ramekins, cover lightly with plastic wrap, and chill overnight. If you don’t plan to eat it right away, cover with a thin layer of melted butter to seal, cool until hardened, then wrap tightly in plastic wrap and refrigerate for up to 1 week or freeze for up to 2 months.