Melt the butter in a small saucepan over medium heat.
Cook, swirling occasionally, until it turns amber and smells nutty, about 4 minutes. Remove from heat and let cool for 2 minutes.
Combine the brown butter and bourbon in a jar and stir well. Let it sit at room temperature for 30 minutes, then freeze for at least 2 hours, until the fat solidifies on top.
Skim off and discard the solidified fat.
Strain the bourbon through a coffee filter or cheesecloth to remove any remaining milk solids. This yields brown butter bourbon.
Combine 2 ounces of the brown butter bourbon, the maple syrup, Angostura bitters, and orange bitters in a mixing glass with ice. Stir for 30 seconds, until well chilled.
Hold one end of the cinnamon stick with tongs and light the other end with a lighter or match until it smolders and smokes (it won't sustain a flame on its own).
Take a heatproof rocks glass made out of thick glass, and invert it over the smoking end to trap the smoke inside. Let it sit for 15 to 20 seconds.
Lift the glass away, top it with a coaster to keep the smoke trapped, and set it right-side up.
Remove the coaster, add a large ice cube, and pour the cocktail over it.
Express the orange peel over the drink and drop it in.
Garnish with the burnt cinnamon stick and serve the cocktail.