Place bread flour, sugar, yeast, and salt in the bowl of a stand mixer. Mix with a fork. Add milk, egg yolks, and butter. Attach dough hook and knead until dough forms a ball and feels smooth and elastic, about 10 minutes.

Pour vegetable oil into a large bowl. Add dough and roll around to coat with oil. Cover with a damp cloth and let rise in a warm place until doubled, 1 to 2 hours.

Transfer dough to a flat work surface. Knead gently to knock the air out. Form dough into a flat oblong shape, then roll into a log. Cut the dough into 4 equal pieces. Keep dough covered with a damp cloth while you work with 1 piece at a time.

Dust work surface with a little flour. Use a rolling pin to roll out first piece of dough. Invert a round 8-inch pan or plate on top; trace the rim with a sharp knife to cut dough into an 8-inch circle.

Line a baking sheet with parchment paper and slide the circle onto it. Warm cookie butter briefly in the microwave to make it easier to spread. Spread 1/3 of the cookie butter over the dough circle.

Repeat with 2 more pieces of dough and remaining cookie butter. Roll out last piece of dough, cut into an 8-inch circle, and place on top. Invert the pan or plate over the stack and cut through all the layers to make a perfect circle.

Invert a small drinking class in the center of the circle. Use a sharp knife to cut dough into quarters, from the base of the drinking glass to the outside edge. Cut quarters into eighths, then into sixteenths.

Grab 2 slices and twist each one twice, in opposite directions. Repeat all the way around the circle. Pinch and seal all the ends. Cover with a damp cloth and let rise for 20 minutes.

Preheat the oven to 350 degrees F (175 degrees C). Beat egg whites lightly and brush over dough.

Bake in the preheated oven until golden brown, about 20 minutes. Cool briefly before serving.