Preheat oven to 350 F. Line a 1-pound loaf pan with unbleached parchment paper on all sides, set aside.
In a small saucepan, brown your one stick of butter by melting it, it will break down into separate liquid fat and milk solids. The milk solids are what will become toasted brown, make sure to stir often to move around the milk solids so they don't burn. Pour into a small bowl and let cool in the fridge until it’s soft and partially solidified- if you wait too long, it becomes hard and difficult to mix.
While waiting make cinnamon sugar apples by peeling, coring, and dicing 2 medium apples. In a small bowl mix diced apples, 1 tsp cinnamon, and 2 tbsp brown sugar until well coated. Set aside.
We want to make our streusel topping ahead of time so you’re not rushing around before getting it in the oven. In a small bowl, simply mix 2 tbsp of softened butter, ⅓ cup of sugar, 1 tsp of cinnamon, and 1 tbsp of flour until combined. Set aside.
In a medium bowl, mix your browned butter, ⅓ cup sugar, ⅓ cup brown sugar until slightly fluffy. Add eggs and vanilla extract until just combined.
In a separate small bowl combine the ¼ cup milk and ⅓ cup greek yogurt. Add half of this mixture into the main bowl with the other ingredients, then set the remainder aside.
Mix 1 ½ cups of flour, 2 ½ tsp of baking powder, and 1 1/2 tsp of salt into the wet ingredients and then fold in the remaining milk/yogurt mixture.
Pour half of the cake batter into the loaf pan and then sprinkle the apples on top. Cover the apples with the remaining cake batter.
Take the streusel topping and separate into dime-size pieces and sprinkle on top. Bake for 50-60 minutes or until golden brown and a knife comes out clean when inserted.