Add the flour, yeast, salt and sugar to the mixing bowl and stir quickly to combine.
Next, add in the eggs, milk and butter. Stir on low until the ingredients come together and then increase the speed to medium. Continue to mix for 12-15 minutes, until the dough is shiny and supple and forms a ball around the dough hook.
Cover the dough and allow it to rise until doubled, about an hour.
Divide the dough into 8 pieces. Press each piece into a rectangle gently with your fingertips, about 2 inches tall aby 5 inches long. Take the bottom edge and fold it up about 2/3rds over. Then fold the top side down about 2/3rds of the way. Fold the left and right sides in about 1/2 of an inch. Finally, fold the top and bottom sides together and pinch to seal. Flip the bun into the prepared pan, seam side down.
If using a hot dog bun pan, place the buns seam side down into the prepared mold. If using a baking dish, place each bun seam side down in the dish about 1/4 of an inch apart, side by side. The buns will puff up and be touching.
Cover again and let the buns rise a second time until puffed up, about 45 minutes.
Bake at 350 degrees for 18-20 minutes, or until the internal temperature is 200 degrees and the buns are golden brown.
Allow the brioche hot dog buns to cool completely on a cooling rack. To serve, take a bread knife and make a slit about halfway through the bun. Enjoy with all the fixings!