Melt the butter in a skillet over medium heat. Add the diced apples, brown sugar and cinnamon and cook for 4-5 minutes, stirring often, until the apples are just tender and the juices turn syrupy. Spread them on a plate and refrigerate until completely cold, about 15 minutes.
Preheat the oven to 425°F and line a baking sheet with parchment paper. Blend the cottage cheese and vanilla until completely smooth and refrigerate until needed.
Whisk the flour, sugar, baking powder, baking soda, salt and cinnamon in a large bowl. Grate the frozen butter into the flour on the large holes of a box grater and toss to coat.
Add the cold apples and toss them through the flour mixture. Pour in the blended cottage cheese and stir with a fork until a shaggy dough forms, adding milk 1 tablespoon at a time only if dry flour remains.
Turn the dough onto a floured surface and pat it into an 8-inch square about 1 inch thick. Cut it into 9 squares with a bench scraper and place them 2 inches apart on the baking sheet.
Bake for 16-19 minutes, until deep golden and the apples are caramelized at the edges. Let the biscuits cool for 5 minutes.
Whisk the powdered sugar, milk and vanilla into a smooth, pourable glaze. Dip the tops of the warm biscuits into the glaze or spoon it over them, and let it set for 10 minutes before serving.