Preheat oven to 375°F (190°C). Line a 9x5-inch loaf pan with parchment paper and set aside.
In a small saucepan, brown the butter over medium heat until golden brown with a nutty aroma. Watch the butter carefully, as it can easily go from browned to burnt! Set aside to cool.
If you wish to peel the apples, do so before chopping. Chop apples into small pieces and place in a mixing bowl.
In a bowl, combine melted butter, brown sugar, cinnamon, and flour. Use a fork to mix it until it is crumbly. Set aside.
In a large bowl, whisk together flour (removing 2 teaspoons of flour and adding to the bowl of chopped apples), baking powder, salt, and cinnamon. In a separate bowl, whisk the browned butter, sugar, eggs, Greek yogurt, and vanilla until smooth. Fold the wet ingredients into the dry.
Mix the flour and chopped apple combination until the apples are coated in flour. Gently fold into the batter (adding flour to the apples helps prevent them from sinking to the bottom of the loaf!).
Pour half the batter into the prepared loaf pan, then sprinkle with half the crumble topping. Add the remaining batter on top, followed by the rest of the crumble. Bake for 55–60 minutes, or until a toothpick comes out clean.
Once bread has cooled, whisk together powdered sugar and heavy cream. Drizzle over the loaf before slicing and serving.