Put the garlic cloves in a saucepan with just enough olive oil to cover them. Place over low heat and simmer until the cloves are soft and golden, 30-40 minutes. With a slotted spoon, scoop the garlic into a bowl, then mash to a puree. Immediately strain the oil through a coffee filter and reserve.
Process the butter in a food processor until smooth and creamy. Add 1/4-cup of the garlic puree (save any extra for spreading on bread), 1-1/2 Tbs. of the reserved garlic oil, the thyme, and a pinch of salt. Process until well blended. Taste and add more salt if needed. Refrigerate until firm enough to shape into a log.
Put an 18-inch sheet of aluminum foil on your work surface. Spoon the butter down the center of the foil into a log about 1-1/2 inches in diameter. Enclose in foil and twist the ends to make a sealed log, like a Tootsie Roll. Refrigerate for up to 1 week, or freeze for up to 6 months.
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