Preheat the oven to 350°F (175°C) and lightly grease a 6-hole donut pan with extra butter; set aside.

In a medium bowl, whisk together the flour, baking powder, ground cinnamon, and salt. Set aside.

In a large bowl, use an electric mixer to beat the room-temperature butter with the brown sugar and 2 tablespoons granulated sugar until creamy, about 1–2 minutes.

Add the egg and vanilla. Beat until combined.

Mix in the applesauce and milk; beat just until combined.

Add the flour mixture and beat briefly until the batter is smooth and homogeneous. Scrape the bowl once to ensure even mixing — do not overmix.

Spoon the batter into a large piping bag (or a resealable plastic bag with the corner snipped) and pipe the batter into the donut cavities about 3/4 full. Alternatively, use two spoons to fill each cavity.

Bake for 18–22 minutes, or until a toothpick inserted into the center comes out clean and the donuts are set and lightly golden.

Remove the pan from the oven and cool for 5 minutes. Gently remove the donuts from the pan and transfer to a wire rack.

In a small bowl, combine 1/3 cup granulated sugar with 1/2 tsp cinnamon.

Brush each warm donut lightly with the melted butter, then roll or dip in the cinnamon-sugar mixture to coat.