Special equipment: One 12-cup or two 6-cup doughnut pans, preferably nonstick; a pastry bag
For the doughnuts: Preheat the oven to 375 degrees F. Spray a 12-cup doughnut pan with nonstick spray.
Put the butter in a small saucepan over medium heat. Cook, swirling occasionally, until golden brown, about 6 minutes. Transfer the brown butter to a measuring cup and set aside.
Combine the flour, baking powder, baking soda, salt and nutmeg in a medium bowl. In a large bowl, whisk together the sugar and eggs until smooth; then whisk in the buttermilk and vanilla until combined. Add the flour mixture and half of the brown butter (1/4 cup) to the egg mixture; mix until just combined -- DO NOT OVERMIX.
Transfer the batter to a pastry bag and fill the doughnut pan about three-quarters full. Bake the doughnuts until light golden brown and just baked through, about 8 minutes; (They will be darker on the bottom, where the batter cooks against the pan, than on the top.) Set the pan on a baking rack and let cool for 10 minutes. Remove the doughnuts from the pan and place them directly on the baking rack. Set the rack over some parchment or foil before glazing.
For the glaze: Bring the maple syrup and the remaining half of the brown butter (1/4 cup) to a simmer in a small saucepan over medium heat. Add the bourbon, cook for 1 minute, and then remove from the heat and whisk in the confectioners' sugar until smooth and slightly thick. Transfer the glaze to a shallow medium bowl and leave to thicken slightly. Drizzle the glaze over the doughnuts, sprinkle the pecans on top, and serve.