In a small saucepan, add butter. Allow to melt and swirl the pan around or stir with a spatula until butter takes on a darker brown colour and small specks form in the bottom. Remove and set aside to cool - don't allow to burn!
Preheat oven to 160ºC. Line an 18cm springform cake tin with baking paper.
In a large bowl, whisk together cooled brown butter, brown sugar, eggs, Greek yoghurt and vinegar until well combined.
To the wet ingredients, add flour, salt and cinnamon and fold through with a spatula until no dry parts remain. Dice apples into chunks (roughly one centimetre pieces). Add to batter and fold in gently.
To make the crumble: combine all ingredients in a bowl. Using your finger, rub the dry ingredients into the softened butter until the mixture resembles wet sand. Some larger chunks are fine.
Transfer cake batter into prepared tin and sprinkle the crumble mixture on top.
Bake cake for 40-45 minutes, or until the crumble is golden brown and an inserted skewer comes out clean. Allow to cool in the cake tin for 5-10 minutes, before removing.
Serve you Apple Crumble Tea Cake warm or at room temperature, drizzled or dolloped with cream and a sprinkle of icing sugar.