Heat the oven to 350°F (180°C). Grease a 12-cavity donut pan well, including the center posts.
Combine the all-purpose flour, brown sugar, and cinnamon. Rub in the cold butter with your fingertips or work it in with a fork until coarse crumbs form. Stop mixing as soon as the mixture is crumbly, then set it aside.
Whisk the self-rising flour, granulated sugar, cinnamon, and mixed spice in a large bowl. In a second bowl, whisk the whole milk, oil, and eggs. Add the wet mixture to the dry ingredients and whisk only until no dry flour remains; do not overmix.
Toss the finely diced apples with the 1 teaspoon self-rising flour, then fold them into the batter. Transfer the batter to a piping bag or a large zip-top bag with one corner snipped. Pipe the batter into the prepared pan, filling each cavity about two-thirds full.
Divide the crumble evenly over the batter. Bake for about 15 minutes, then check the cake portion with a toothpick, avoiding the apple pieces. If the donuts are set but the crumble is still pale, increase the oven to 375°F (190°C) and bake for up to 5 minutes more. Cool in the pan for 5 minutes, then transfer the donuts to a wire rack.
Whisk the powdered sugar and cinnamon with 1 tablespoon water. Add more water a little at a time only if needed to make a thick, pourable drizzle. Drizzle over completely cooled donuts.