Prepare the brown butter by melting the butter in a small saucepan over medium heat. Watch carefully as it foams, then turns golden and develops a nutty aroma (about 3-4 minutes). Remove from heat immediately when you see brown bits forming at the bottom. Strain through a fine mesh sieve and let cool slightly.
Make the chestnut base by whisking together the chestnut purée, maple syrup, vanilla extract, cinnamon, and nutmeg in a small bowl until smooth. Divide this mixture between two coffee mugs.
Brew your espresso shots directly over the chestnut mixture in each mug. Stir well to combine and dissolve the chestnut paste completely.
Create the brown butter foam by combining the slightly cooled brown butter (still liquid but not hot) with the half-and-half and powdered sugar in a tall container. Froth using a milk frother or immersion blender until light and airy with soft peaks.
Assemble your drink by gently spooning the brown butter foam over the espresso mixture, allowing it to create a distinct layer on top.
Garnish with a sprinkle of cinnamon, a few small pieces of chestnuts, or a cinnamon stick if desired. Serve immediately while the contrast between the hot espresso and creamy foam is at its peak.