Preheat oven to 375 degrees F. Grease and flour two 12 (3-inch) madeleine pans. Set aside. (If you only have one, you can bake the cookies off in batches, just re-grease and re-flour after each batch).
Melt the butter in a medium saucepan, over medium heat until the crackling stops and brown bits form at the bottom. Immediately remove from the heat and pour out into a small dish, scraping up all those bits from the bottom. Set aside and cool to room temperature.
In the bowl of your stand mixer, fitted with the paddle attachment, beat the eggs, vanilla and salt, at high speed until light and fluffy, about 8 minutes. While the machine is running, slowly pour in the sugar in a slow and steady stream. Continue to be until the mixture is pale and very thick, 5 to 10 minutes.
Add the flour into the mixture in thirds, gently folding the flour in after each addition. Make sure not to over-mix the batter at this point or you'll end up with dry madeleines.
Stir in the orange zest, nutmeg and browned butter, make sure to gently fold it in, as quick as possible. Spoon the batter into the molds and give the pans a gently rap on the counter to dislodge any air bubbles. Bake until golden brown around the edges and the cookies spring back when you touch them, about 12 to 14 minutes. Remove from the oven and let them cool in the pans for about 5 minutes. Use a pairing knife to loosen from the pan and then invert onto a wire rack.
To make the chocolate sauce, add cream to a medium saucepan along with the brown sugar, cocoa powder, espresso powder, and salt and cook over medium-high heat, whisking occasionally, until sugar is dissolved, no lumps of cocoa powder remain, and mixture is simmering, about 4 minutes. Remove from heat and add chocolate, stirring to melt. Keep warm over very low heat until ready to serve.
In a medium bowl, toss together the sugar and cinnamon until well combined. Once the cookies have cooled, dip each into melted butter and then roll around in the cinnamon sugar mixture. Serve immediately and enjoy!