Warm your milk. You do not want it hot. I find that about 1 minute in the microwave is sufficient.
Add the milk, sugar and sourdough start to the bowl of a stand mixer.
Give it a quick mix to incorporate.
Add your flour and eggs to the mixer and mix it into a shaggy dough on medium speed. Scrape down the sides of the mixing bowl.
Let it sit for about 20-30 minutes.
After this you want to add the salt and butter. You will be adding the butter in small amounts. You can add about 1 tbsp at a time. Make sure to use your dough hook attachment for this.
Once all the butter and salt is mixed in, let it knead in the mixer until the dough is elastic and soft. It should pull away from the sides of bowl. It should feel silky soft. It shouldnt feel very sticky, but should be really soft. If its too sticky add a small amount of flour and continue to knead. Kneading helps to develop the gluten in the dough. Move the dough into a container or bowl so that it can bulk ferment. Cover and set it in a warm spot. This will help it to rise a bit faster.
It will take some time so be patient and do other things while your dough do its thing.
Once the dough has risen to at least 50 percent its time to shape it and put it in your loaf pan. This could take from 6-8 hours.I actually prefer to mix this dough at night and let it sit on my counter overnight. It will be doubled by morning. In the morning, the dough will be ready to be shaped into buns. I use 50 grams of dough for rolls. You can separate and weigh out your dough.
One it is weighed out, you can shape them into balls. Take a piece of dough and mold it into a rough circle. Place it on the counter and use your hand to shape it by moving it in a circle using one hand, creating surface tension on the dough. Place them on a large baking sheet covered in parchment paper. Cover in plastic wrap and let them rise until doubled in size. This may take a few hours. You can place them in the oven uner a proofing setting if you prefer to speed up the process. When they are close to being proofed, preheat the oven to 350 degrees.
Once they have risen, use a brush to apply the egg wash to the top of the buns.
Bake in a preheated 350 degree oven for 30-35 minutes. You want the tops to be golden brown. They should puff up nicely.
Remove your buns from the oven and let them cool.
Cut your buns and add all the toppings you want!
Store in an airtight container.