These are my new favorite things. Savory doughnuts, easily baked, and full of spicy goodness. The texture stays light and fluffy, so you can eat way too many of these before you realize you're a brunch monster. I might even try making these with cornmeal, as sort of an epic breakfast cornbread situation. But until then, nom nom nom nom.
Place a rack in the upper third of the oven and preheat to 350 degrees F. Lightly spray a 6-cavity doughnut pan with nonstick cooking spray and sprinkle with 1/4 cup of the cheese; set aside.
Melt the butter in a medium skillet over medium heat. Continue to cook until the butter is browned and smells nutty, about 5 minutes. Pour the browned butter immediately into a medium heatproof bowl and let cool.
In another medium bowl, whisk together the flour, sugar, baking powder, baking soda, salt and nutmeg; set aside.
Add the egg, buttermilk, scallions and jalapeno to the bowl with the browned butter; whisk together to incorporate. Pour into the bowl of dry ingredients, add the remaining 1/4 cup cheese and stir together just until everything is coated.
Scoop the batter with a small spoon and another small spoon to push the batter into the prepared pan. Fill each doughnut in the pan three-quarters full with batter then smooth with a rubber spatula. Bake until a toothpick inserted into a doughnut comes out clean, 12 to 13 minutes. (The doughnuts won't look super golden brown at this point.) Remove the pan from the oven and immediately flip the doughnuts out onto a rimmed baking sheet.
For crispier, extra golden-brown doughnuts (optional): Arrange the doughnuts on the baking sheet cheesy-side up, return them to the oven and switch the oven to broil. The cheese will bubble a bit and the doughnuts will get a little crispy, about 1 minute. Keep a close eye on the doughnuts, since they'll burn easily.