Put the butter in a small pot over medium-low heat. Cook, stirring occasionally, until it melts, foams, and smells nutty, about 5 minutes. It will suddenly start darkening. When the melted butter turns golden brown, remove the pot from the heat and stop stirring; let the darker solids settle to the bottom.
Heat the water or cider on the stove over medium-high heat until it just comes to a boil. (Or use the microwave.)
Put as much bourbon as you like (if you’re using it) in the bottom of 2 heatproof glasses or mugs. Sweeten the bubbling water (or cider) with some sugar if you’d like. Pour the hot liquid into the mugs. Add a cherry with some juice from the jar. Top the drink with a couple teaspoons of browned butter. You can stir it in if you’d like or sip the drink through the buttery layer floating on top.