Toast the Serrano chile and the shallot in a dry cast iron skillet until they blacken on all sides, about 10 minutes (the shallot may need to be taken out before the Serrano). Allow the chile to cool, and remove the peel (it should pretty much just slip right off. But, if there are little parts with peel still on them, don't worry, it'll be fine.) Remove the stem end from the chile and cut it in half lengthwise. Remove the seeds and the white membranes from inside the chile.
Place the roasted chile, the roasted shallot, salt, olive oil, and butter in the bowl of a food processor and pulse until all ingredients are chopped and well combined.
Let the butter stand, covered, in a cool place for 1 hour to allow the flavors to develop. Store in an airtight container in the refrigerator.
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