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In the bowl of a stand mixer or in a large mixing bowl, combine the starter, water and 1/2 cup of the bread flour to form a thick batter. Cover the bowl and allow the mixture to rise for 30-60 minutes.
Warm the milk in the microwave to scalding. Add the butter to the milk and stir until the butter melts. Allow the milk to cool to 110°F. Add the milk, sugar, salt and egg to the sponge. Mix to combine.
Add 1 1/2 cups of the bread flour. Mix until the batter looks like thick pancake batter. If using a stand mixer, switch to the dough hook.
With the mixer running, slowly add the remaining cup of the bread flour. Mix until the dough begins to gather on the hook and clears the sides of the bowl. Knead for 5 minutes. If mixing by hand, add as much of the flour as you can in the bowl then finish kneading in the rest of the flour by hand And then knead 5 minutes.
Dump the dough onto a lightly floured surface. The dough should be soft and slightly sticky. Knead to form a smooth ball. If the dough is very sticky sprinkle a little more flour as you knead. Place the dough in an oiled bowl, turning once to coat the dough. Cover and set aside in a warm spot.
After 30 minutes uncover the bowl, lift one side of the dough and fold it into the middle of the dough. Repeat with the other three sides of the dough then flip the dough over. You're basically turning the dough inside-out to redistribute the yeast and strengthen the gluten. Cover the bowl and after 30 minutes repeat the procedure. Cover the bowl and after 60 minutes repeat the procedure again.
Cover the bowl and after 60 minutes fold the dough one more time. By now the dough should be lively, elastic and airy. If the dough is still sluggish give it another hour or two at room temperature. If you want to finish making the buns the next day return the dough to the bowl, cover tightly and refrigerate overnight. The next morning continue with shaping. Otherwise continue shaping the buns on the same day.
Turn the dough out onto a lightly floured surface and divide the dough into 6 or 8 equal portions (based on how big your burgers are). Roll each portion to a ball and set them a parchment lined baking sheet. Use the palm of your hand to flatten each ball to a disc about 1" thick.
Cover the sheet pan and allow the rolls to rise until doubled in size (about 1 hour, longer if the dough is cold). Press a roll with your finger. If it springs right back it's not ready, if the dent slowly fills in the rolls are ready to bake.
Meanwhile, preheat the oven to 400°F.
Brush the tops of the buns with egg wash. Sprinkle each bun generously with sesame seeds. Bake until the buns are golden brown and feel light when lifted off the sheet pan, about 10-15 minutes. Allow to cool completely before slicing.