Dough Combine flour, sugar, yeast, and salt. Cut cold butter into flour mixture.
In a separate bowl, whisk egg, milk, and vanilla; pour into flour mixture; mix just enough to form a dough.
Transfer dough to a clean dry surface; shape the dough into a rectangle; wrap in plastic wrap; chill at least 3 hours, or overnight. Place chilled dough on a floured surface; roll into a rectangle, ~9×15 inches; fold dough in thirds, like a letter; quarter-turn this layered dough; roll out again; fold into thirds; repeat this process two times; wrap dough in plastic wrap and chill ~30 minutes (if it warms and begins to stick to the counter).
Combine white sugar, brown sugar, cinnamon, and salt; coat a 12-well muffin tin with melted butter or non-stick cooking spray. Roll dough into a rectangle ~12×18 inches; coat the dough with butter; leaving ~1-inch clear at far end.
Starting at the wider side, tightly roll dough into a log.
With a serrated knife, cut the roll into 12 pieces; place each in a muffin well.
Cover the dough and let rise ~one hour, until doubled in size.
Preheat oven to 375°F; uncover pan; bake ~20 minutes, until golden brown.
Let cool in the pan ~10 minutes; then place each bun on a wire rack to cool completely.
