Preheat the oven to 400°F and line a baking sheet with parchment paper. Blend all of the cottage cheese (both the biscuit and filling amounts) until completely smooth. Transfer 3/4 cup to a bowl for the filling, stir the vanilla for the biscuits into the rest and refrigerate it.
Whisk the reserved 3/4 cup blended cottage cheese with the sugar, egg yolk, cornstarch, lemon zest and vanilla until smooth. Refrigerate the cheesecake filling while you make the dough.
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.
Pour in the vanilla cottage cheese and stir with a fork until shaggy, adding milk 1 tablespoon at a time only if needed. Turn the dough onto a floured surface, pat it out, fold it in half twice, then pat it to 3/4 inch thick.
Cut out 3-inch rounds and place them 2 inches apart on the baking sheet. Dip the bottom of a small glass in flour and press a deep well into the center of each biscuit, leaving a 1/2-inch rim.
Spoon a heaping tablespoon of cheesecake filling into each well and top with a teaspoon of raspberry jam, swirling it lightly with a toothpick. Brush the rims with milk and sprinkle them with coarse sugar.
Bake for 16-19 minutes, until the biscuits are golden and the filling is set with only a slight wobble. Let them cool on the pan for 15 minutes so the filling firms up, then dust with powdered sugar and serve.