Combine the butter, bourbon, maple syrup, brown sugar, and ⅛ teaspoon of the salt in a small bowl. Using a fork, mash until well combined. Alternatively, pulse the ingredients together in a food processor. Taste and season with additional salt as needed. Scrape the butter into a ramekin and cover with plastic wrap. Alternatively, scrape the butter onto a sheet of parchment paper and form into a log about 2 inches in diameter, rolling the parchment paper to cover the log and twisting the ends of the paper to close. Wrap in foil to keep airtight. Chill in the refrigerator or freeze.