Using the beater, barely combine first 4 ingredients. Cover and set aside for 15 min. Add butter, brown sugar, eggs, salt and vanilla and mix well. 

Still using the beater attachment, add 2 cups of flour. Then begin adding flour 1/4 cup at a time. This dough will be a little stickier than normal bread dough, but the desired consistency is to still be able to handle it easily. 

Beat for 3 min. Remove beater attachment and add bread hook. Beat for 5 min.

Place in a lightly greased (8 cup) bowl and cover to allow the dough to rise. Let it set for 1.5 hours, or even 2+ hours, if necessary, letting the dough double in size. (Sweet dough tends to take longer to rise)

Knead the dough.

Generously grease a 9×13 pan. Divide dough into 10 equal portions. (If weighing dough, use 3.1 oz of dough per bun) Roll each portion into about a 5″ long, skinny rope. Leaving a little space between each one, place the future bun in neat rows. 

Allow to rise for another hour. Bake at 350°F for 20-25 min. Using the reserve egg white, with 5 min. left to bake, baste the tops of the buns.

Turn out, while hot, onto cooling rack.

Yields 10 hot dog buns.