• Preheat the oven to 350°F (175°C). Grease an 8-inch square pan and line it with parchment paper, leaving a little overhang on two sides so you can lift the cake out easily later.

Brown the butter

• In a light-colored saucepan over medium heat, melt the butter, swirling the pan often. It will foam and sputter, then quiet down. Keep swirling until the milk solids at the bottom turn a deep golden brown and the butter smells nutty and toasty, 5 to 7 minutes.
• Immediately pour the brown butter into a large heatproof bowl, scraping in every last brown speck (that's where all the flavor lives). Set aside to cool while you make the filling.

Apple pie filling

• In a large skillet over medium heat, melt the butter. Add the apples, brown sugar, cinnamon, nutmeg, allspice and salt. Cook, stirring often, until the apples are just tender but still hold their shape, 7 minutes.
• Meanwhile, in a small bowl, stir together the cornstarch and lemon juice until smooth. Once the apples have cooked for 7 minutes, pour the mixture over them and stir to combine. Let it simmer for 2 more minutes, until the juices thicken and turn glossy. Spread the filling onto a plate and let it cool completely.

Crumb topping

• In a small bowl, stir together the flour, brown sugar, granulated sugar, cinnamon and salt. Pour in the melted butter and mix until no dry flour remains. At this point it'll look more like a soft dough than a crumble, and that's exactly what we want.
• Transfer the bowl to the refrigerator while you make the batter. As the butter firms up, the mixture will be ready to break into crumbles of every size.

Batter

• In a large bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg and cloves.
• Add all of the wet ingredients right on top: the cooled brown butter, avocado oil, brown sugar, eggs and egg yolk, sour cream and vanilla. Whisk everything together, then switch to a spatula and gently fold just until no streaks of flour remain. The batter will be thick.

Assemble and bake

• Spread half of the batter into the prepared pan in an even layer. Spoon half of the apple filling over the top.
• Dollop the remaining batter over the apples and gently spread it to the edges. Dolloping first makes it much easier to spread without dragging the apples up into the batter, and it's completely fine if a few peek through.
• Scatter the remaining apples over the top. Take the crumb topping out of the fridge and break it apart with your fingers into a mix of big, chunky clusters and tiny crumbs, then sprinkle it evenly over the cake. Use all of it!
• Bake until a toothpick inserted into the center comes out clean or with a few moist crumbs, 65 to 70 minutes. If the topping is browning too quickly, tent it loosely with foil.

Glaze

• While the cake bakes, whisk together the powdered sugar, 2 tablespoons of milk, the vanilla and the salt in a small bowl. Add more milk a teaspoon at a time until the glaze is smooth and pourable, but not watery.

Back to our cake

• Let the cake cool in the pan for at least 30 minutes, then use the parchment overhang to lift it onto a wire rack.
• Drizzle the glaze over the cooled cake and let it set for about 10 minutes before slicing and serving.