Brown the butter. Add the butter to a small saucepan over medium heat. Stir frequently as it melts. Continue cooking until the milk solids turn golden brown and the butter smells rich and nutty, about 5–7 minutes. Remove from the heat and let cool for 2–3 minutes.
Combine with the bourbon. Pour the browned butter and bourbon into a large glass jar (such as a Weck jar). Seal tightly and shake to combine.
Infuse. Let the mixture sit at room temperature for 2–4 hours, giving the jar a shake every 30 minutes or so.
Freeze overnight. Transfer the jar to the freezer for 8–12 hours, or until the butter has hardened into a solid layer on top.
Remove the butter. Lift off the solid butter cap and discard it, or save it for cooking or baking (it'll have a subtle bourbon flavor).
Strain. Pour the bourbon through a fine-mesh strainer lined with cheesecloth or a coffee filter to remove any remaining butter solids. For an extra-clear result, strain a second time through a fresh coffee filter.
Bottle and store. Funnel the finished bourbon into a clean glass bottle or jar.