Hamdog bun: in a large bowl or the bowl of your mixer, weigh out and add your all-purpose flour, sugar, salt, and instant yeast and stir to combine all your dry ingredients to ensure no clumps. 

Add the butter and room temperature water to the dry ingredients. 

If using a stand mixer, knead for 8 minutes on medium speed. If you are kneading by hand, you will need about 15 minutes of kneading after all the ingredients are incorporated together. 

The dough should be smooth and a little bit sticky. 

Place your dough in a lightly oiled bowl and cover. Let dough rise in a warm place for 1 to 1.5 hours. The dough should be doubled in size. 

Remove dough from the bowl to a lightly oiled counter and divide into 5 balls that weigh 65 grams and 5 "log" shaped dough pieces that weigh 45 grams. There will be some dough leftover that you can form into a regular-shaped bun. 

Flatten the larger balls of dough into flat circles. Then flatten the middle part of the dough logs and lay the circle in the center of the log. Using your fingers, pinch together the edges of the two dough pieces until they form a more consistent hamdog-shaped piece of dough. 
 

Add each shaped hamdog bun to a parchment-lined sheet pan. 

Cover your buns with lightly greased plastic wrap or a second sheet pan flipped over on top of the pan you're using. 

Let the shaped buns rise for another hour or hour and a half until they are nearly doubled in size.

Near the end of your final proofing time, preheat your oven to 375 degrees F (177 C). 

Add a whole egg white to a bowl with a tablespoon of water. Whisk until well combined. Carefully paint this egg white wash on top of each hamdog bun and then sprinkle liberally with sesame seeds. The egg white wash will help the seeds stick to the top of the buns and will help form a nice crust. 

Bake your buns for 15 to 20 minutes rotating the sheet pan halfway through the baking time. I typically bake for 7 minutes and then rotate and come back in another 7 minutes to see what my level of brownness looks like. At that point your buns should be done all the way through, you're just trying to get them to the color you prefer. 

Remove the hamdog buns from the oven and allow them to cool before slicing.

Meat cooking: place a medium skillet or griddle over medium-high heat. Add a hot dog to the pan.

Form the quarter pound of ground beef into a thin patty (about a 3/4-inch thick) that is a little bit wider than the round part of your hamdog bun (or as close as you can get). Salt and pepper the ground beef patty and place the seasoned side down in the hot pan. Season the second, top, side, and allow the patty to cook, untouched, for 3 to 4 minutes.

Flip the patty and cook for 1 minute. Using a spatula, cut the burger patty directly down the middle of the patty. Using tongs or a fork, move the hot dog into the center of the patty. Top the burger patty and hot dog with a slice of cheddar cheese (or your favorite cheese) cook the second side for at least 2 minutes. While this second side is cooking, add hot dog chili to the top of the cheese. 

Remove the patty to a plate while you assemble the rest of your hamdog.

Hamdog assembly: slice and toast your bun if desired. Add any optional condiments to the bottom bun.

Carefully add the hamdog patty topped with cheese and chili to the bun. 

Top with pickled Vidalia onions and close the hamdog with the top of the bun.

Serve and enjoy.