Prep the apples. Peel if you prefer a smoother texture, or leave the peel on for a rustic look. Grate the apples on the large holes of a box grater. Toss with lemon juice. If very juicy, lightly squeeze a handful to remove excess, but keep most of the moisture.

Mix dry ingredients. In a large bowl, whisk flour, baking powder, baking soda, salt, cinnamon, and nutmeg until well combined.

Mix wet ingredients. In a separate bowl, whisk eggs, both sugars, melted butter, buttermilk, and vanilla until smooth and slightly thickened.

Combine. Stir the wet mixture into the dry with a spatula until just combined. Fold in the grated apples. The batter should be thick but scoopable. Avoid overmixing.

For baked donuts: Preheat oven to 350°F (175°C). Grease a donut pan well. Spoon or pipe batter into the molds, filling each about 3/4 full. Bake 12–15 minutes, until the tops spring back and a toothpick comes out clean. Cool 5 minutes in the pan, then transfer to a rack.

For fried donuts: Heat oil to 350°F (175°C) in a heavy pot. Lightly flour your hands and a surface. Turn the batter onto the floured surface—it will be soft. Gently pat to 3/4-inch thickness with floured hands. Cut with a donut cutter or use a large and small round cutter. Fry 1–2 minutes per side until deep golden. Drain on a rack set over a sheet pan.

Add a finishing touch. While warm, toss baked or fried donuts in cinnamon sugar, dust with powdered sugar, or dip in vanilla glaze. For cinnamon sugar, mix 1/2 cup sugar with 1 teaspoon cinnamon. For glaze, whisk 1 cup powdered sugar with 2–3 tablespoons milk and 1/2 teaspoon vanilla.

Let them set. Give glazed donuts 15 minutes to set. Cinnamon sugar can be added right away while the donuts are still warm.