Pour the apple cider into a small saucepan and simmer over medium heat until reduced to 1/2 cup, about 15 to 20 minutes. Set aside to cool slightly.

Melt the butter in a light-colored skillet or saucepan over medium heat. Continue cooking until golden brown and nutty smelling, then remove from the heat and let cool slightly.

In a large bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.

In a medium bowl, whisk together the sugar, eggs, reduced cider, milk, vanilla, and browned butter until smooth.

Add the wet ingredients to the dry ingredients and stir just until a soft dough comes together.

Turn the dough onto a lightly floured surface and gently pat or roll it to about 1/2-inch thickness. Cut out 12 ring donuts, re-rolling scraps as needed.

Heat about 2 inches of neutral oil in a heavy pot to 350°F. Line a tray with paper towels and mix the sugar and cinnamon for the coating in a shallow bowl.

Fry the donuts in batches for 1 to 2 minutes per side until deep golden brown and cooked through, adjusting the heat as needed to keep the oil near 350°F.

Transfer the donuts to the paper towel-lined tray to drain briefly, then roll them in the cinnamon sugar while still warm.

Serve warm or let cool slightly before storing.