Pulse butter, cheddar, rosemary, flour, salt, and cayenne in a food processor until mixture resembles cornmeal. Add vinegar and 2 tbsp. very cold water and pulse just until mixture holds together. Lay a piece of plastic wrap on your counter and turn dough out onto it. Form dough into a 6-in. square, cover with more plastic wrap, and chill at least 1 hour.
Preheat oven to 400°. Put chilled dough on a lightly floured surface and roll out into a 12-in. square. Cut square into 48 1- by 3-in. strips. Wrap each date with a pastry strip, then arrange on 2 baking sheets. Bake until pastry is light golden, about 15 minutes. Serve hot.
Note: Nutritional analysis is per serving.