Preheat the oven to 350 degrees F. Generously butter a 12-cup madeleine pan, making sure to get into all the grooves. Dust with flour and tap out the excess.
Melt the butter in a small skillet over medium heat, swirling the skillet occasionally, until brown flecks appear and the butter smells nutty, 3 to 5 minutes. Set aside to cool.
Combine the flour, granulated sugar, lemon zest, nutmeg and salt in a large bowl. Add the eggs and whisk to moisten (they will not fully blend with the flour), then whisk in the cooled brown butter until smooth and shiny.
Spoon 1 tablespoon of batter into each cup of the prepared pan. Bake until the edges are golden brown and the cookies gently spring back when pressed, 12 to 14 minutes. Immediately loosen the edges with a small offset spatula or butter knife and unmold; transfer to a rack to cool completely. Run the pan under cool water, wipe clean and generously butter and flour again. Repeat with the remaining batter. Dust the madeleines with confectioners' sugar.