Preheat the oven to 350°F (175°C) and grease two standard 6-cavity donut pans. Be sure to coat each cavity thoroughly so the donuts release cleanly after baking. Allow the pans to sit for a minute so the grease settles evenly into all the ridges.
In a small bowl, combine the 1/2 cup flour, 1/3 cup brown sugar, and 1/4 tsp cinnamon. Add the cold, cubed butter and work it into the mixture using your fingertips or a pastry cutter. Continue until the mixture forms coarse, crumbly pieces that hold together when lightly pressed. This texture helps the topping stay crisp while baking. Set the bowl aside to keep the butter cold until needed.
In a large bowl, whisk together the 2 cups flour, 3/4 cup brown sugar, baking powder, baking soda, 1 1/2 tsp cinnamon, nutmeg, and salt. Break up any lumps in the sugar as you whisk to ensure an even distribution throughout the dry mixture, which helps the donuts rise uniformly.
In a separate bowl, whisk the egg until its yolk and white are fully combined. Whisk in the buttermilk, melted and cooled butter, and vanilla extract until the mixture becomes smooth and cohesive. Cooling the butter prevents it from solidifying upon contact or cooking the egg prematurely.
Pour the wet ingredients into the bowl of dry ingredients and fold gently with a spatula until just combined. The batter should look thick and slightly lumpy, which is ideal for tender baked donuts. Fold in the finely diced apple, distributing it evenly so each donut contains small pockets of fruit without over-mixing the batter.
