Preheat your oven to 375°F. Line an 8x8 cake pan with parchment and set aside.
Brown your butter. In a small, light colored or stainless sauce pan add your 56g or ¼ cup of butter and place over medium heat. The butter after it melts will foam, then subside, and then bubble loudly. Start stirring after the butter has melted. When the butter turns golden brown and the milk solids are toasted, the butter will have a nutty fragrance. Quickly remove from the heat and into a large bowl.
Let the butter cool for about 10 minutes.
While your butter is cooling, make the topping for the cake. Peel your apple and finely dice.
In a small to medium size bowl, melt your 42g/3tbsp of butter. Add your brown sugar (67g/⅓ cup), cinnamon, and diced apple and stir. Add in your oats and flour and stir well.
Add your oil and brown sugar to your cooled browned butter and stir.
Add in your vanilla and apple cider reduction and mix into the wet ingredients.
Whisk in your egg until it is homogenous.
Add in your buttermilk and combine.
Add your dry ingredients: flour, salt, and baking powder to your wet ingredients. Stir until no dry bits remain, but it is normal/fine if the batter is somewhat lumpy.
Pour your batter into your prepped cake pan and smooth into an even layer.
Spread the apple streusel topping over the cake into an even layer as well.
Bake your cake on the center rack of your oven for about 28-34 minutes. It will be difficult to tell when it's completely done but the topping will look set, the cake will smell fragrant in the oven and a toothpick will come out cleanly.
Remove the cake from the pan and onto a wire rack to cool.