Melt 6 tablespoons of the butter in a heavy pan over medium heat. After it melts, continue to cook the butter, watching carefully. It will foam and subside, then separate into golden butterfat and cloudy white milk solids. The milk solids will begin to brown. When they are lightly browned and the butter smells nutty and toasted, remove from the heat and set aside for 5 minutes to cool. Strain the butter through a fine strainer to remove the larger brown bits.
Whip the eggs and sugars until light and fluffy in a mixer fitted with a whisk attachment. Mixing at low speed, add the dry ingredients. dd the vanilla extract, honey, and brown butter and mix to combine well. (The recipe can be made up to this point and kept refrigerated up to 3 days in advance.)
When ready to bake, melt the remaining 2 tablespoons of butter. Double-butter your baking pans by using a pastry brush to thickly coat the pans with melted butter, then chill in the freezer until firm. Repeat to make a thick coat of cold butter. Heat the oven to 375 degrees F. Pipe or spoon the batter into the madeleine molds, filling them no more than 3/4 full. The recipe can be made up to this point early in the day, then refrigerated until ready to bake. Add 1 minute to the baking time.)
Bake until firm, about 5 to 8 minutes for large madeleines and 5 minutes for miniatures. Immediately knock them out of the pan and serve, or let cool to room temperature, store in an airtight container and serve later the same day.