Preheat the oven to 180°C fan (350°F/gas 4) with an oven shelf in the middle of the oven. Line the base of a 23 cm (9 in) cake tin (pan) with baking parchment or line a 12-cup muffin tray with paper cases.
Wash the apples well, quarter and cut out the seeds. There’s no need to peel the apple.
Add the apples to a high-powered blender or food processor with the cinnamon, sugar, oil and milk and blend until smooth.
Add the flour, baking powder and bicarbonate of soda to a large bowl and gently whisk to ‘sieve’ and combine the ingredients. Add the apple purée and mix well with a whisk or silicone spatula until just combined and there are no dry streaks.
Pour the batter into the lined tin and top with the sliced apple, followed by the Demerara sugar and oat crumble sprinkled across the batter.
Bake for 40 minutes if making the cake or 8–10 minutes for the muffins until golden and the top springs back when gently pressed with your fingertips, or a skewer inserted into the centre of the cake or muffin comes out clean.
Leave to cool completely on a wire rack, then remove from the tin. Serve with a dusting of icing sugar.
This cake keeps very well in an airtight container at room temperature for up to 5 days.