Preparation

1. 1. Put the flours, yeast, sugar, and salt in the bowl of a food processor fitted with the metal chopping blade or dough blade. Process on Low or Dough, if available, to sift. Add the butter and then process again to combine. Combine the water and egg, and while the unit is running on Low or Dough, gradually add the liquid ingredients through the feed tube. Process until the dough cleans the sides of the work bowl and forms a ball. Continue to process to knead for about 30 seconds.
2. Knead by hand on a lightly floured counter for about 1 more minute. The dough should be tender and spring back to the touch.
3. Put the dough into a clean bowl, and cover with a damp towel or plastic wrap. Let rise in a warm, draft-free place for about 90 minutes; the dough should double in size.
4. Divide the dough into 6 pieces. Roll each into a smooth ball. Transfer to a parchment-lined 8-inch round pan, and cover again with a damp towel or plastic wrap. Allow the dough to rise an additional 30 to 40 minutes, or until nicely puffed.
5. Select Bake and set the time to 12 minutes and the temperature to 350°F. Press Start/Stop. Gently flatten the dough balls (this creates a flatter bun, so they do not puff up too much while baking). Brush each with the egg wash, and then sprinkle with the sesame seeds.
6. When the Air Fryer is preheated, carefully place the pan into the basket. Begin checking with 2 minutes left. Rolls should be nicely golden.
7. Remove from basket and allow to cool slightly before halving and serving.