Preheat oven to 350°F. Line an 8″ cake pan with parchment paper and set aside.
In a large bowl or bowl of a stand mixer, use an electric mixer to cream the ½ cup unsalted butter with the 3 tablespoons coconut oil, 2 large eggs, and 1 cup granulated sugar, and 1 teaspoon pure vanilla extract until light and fluffy, about 2 minutes.
On low speed, mix in the 1 ¾ cups oat flour or whole wheat pastry flour, ½ cup almond flour, 1 teaspoon baking powder, and ½ teaspoon fine sea salt. Gently fold in the 2 cups chopped apples and 1 cup blackberries.
Pour the batter into the prepared cake pan, let it rest for 15-20 minutes, then place in the oven and bake for 40-45 minutes, until the top is set and a tester inserted in the middle comes out clean.
Cool the cake in the pan set on a wire rack for 15 minutes, then turn out to cool to room temperature. Transfer to a plate to serve, and sift over some powdered sugar to serve.
Notes
Can I use frozen blackberries? Sure you can, but if you can get your hands on fresh-picked blackberries, please use those! The overwhelming flavor in this cake is blackberry, with the apples playing a bit of a background role, so for the best-tasting berries, use fresh.
Mix in the fruit gently. Use a very gentle hand when you fold in the fruit; the berries especially can get overmixed into the batter. I don't mind a purple cake, but I do like them to keep their shape if possible.
Use another flour. If you prefer, use a 1:1 gluten-free flour for the combination of oat and almond flours (use 2 cups/280 grams of a flour mix, like my homemade gluten free flour blend). Or, use all-purpose flour if you don't have any whole wheat flour on hand.
This cake is a little delicate. Note that the cake is on the delicate side because of all the fruit, so let it cool to room temperature before slicing. Another trick is to keep it in the fridge so the crumb stays nice and firm.
Make it a day ahead. Like many cakes, this apple blackberry cake's flavor deepens overnight. It's delicious on the day it's baked, but I might argue that it's even more so the next day.