Preheat the oven to 425°F and line a baking sheet with parchment paper. Blend the cottage cheese and pumpkin puree until completely smooth, then refrigerate while you prepare the dry ingredients.
Whisk the flour, brown sugar, baking powder, baking soda, salt and pumpkin pie spice in a large bowl, breaking up any brown sugar lumps. Grate the frozen butter into the bowl on the large holes of a box grater and toss to coat.
Pour the pumpkin cottage cheese mixture over the flour and stir with a fork until a shaggy dough forms. Add milk 1 tablespoon at a time only if dry flour remains at the bottom.
Turn the dough onto a floured surface and pat it into a rectangle about 1 inch thick. Fold it into thirds, turn it 90 degrees and pat it out again.
Repeat the folding 2 more times, then pat the dough to an even 1 inch thickness. The folds give these soft pumpkin biscuits flaky, pull-apart layers. Cut out biscuits with a 2 1/2-inch round cutter, pressing straight down without twisting. Place them on the baking sheet so they just touch and chill for 10 minutes.
Bake for 15-18 minutes, until puffed and golden at the edges. Let them cool for 5 minutes.
Whisk the powdered sugar, maple syrup, melted butter and salt until smooth. Spoon the maple glaze over the warm biscuits and let it set for a few minutes before serving.