Preheat the oven to 400°F and line a baking sheet with parchment paper. Blend the cottage cheese, 1/4 cup heavy cream and vanilla until completely smooth, then refrigerate while you prepare the dry ingredients.
Whisk the flour, sugar, baking powder, baking soda and salt in a large bowl. Grate the frozen butter into the flour on the large holes of a box grater and toss to coat.
Stir in the chocolate chips, then pour in the blended cottage cheese mixture. Stir with a fork until a shaggy dough forms, adding 1-2 more tablespoons of cream only if dry flour remains.
Turn the dough onto a floured surface, pat it out, fold it in half and pat it into a circle about 1 inch thick. Cut out 2 1/2-inch rounds, pressing straight down, and press the scraps together once to cut the rest.
Arrange the biscuits 2 inches apart on the baking sheet. Brush the tops with the remaining tablespoon of cream, sprinkle generously with coarse sugar and chill for 10 minutes.
Bake for 16-19 minutes, until golden brown with a crackly sugar top. Cool for 5 minutes and serve warm, while the chocolate is still melty.