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.
You'll never run out of ideas for using this seasoned butter, which you can keep in the freezer and slice as you need. I love it on mashed potatoes and tucked into baked potatoes, or as a finishing butter for risotto, swirled in at the end. You can also toss it with fresh pasta and Parmesan on one of these evenings when you don't feel like cooking, or let a nugget of it melt on a grilled New York steak. Imagine how good a burger would be if the toasted bun were spread with this butter.