To the bowl of a stand mixer add the warm water and sourdough discard. Mix together quickly.
Add in the sugar and instant yeast. Watch for the yeast to activate.
Add the egg, melted butter and salt. Mix again.
With the dough hook running, add the bread flour, reserving about a cup of flour.
Knead for about 5-7 minutes until the dough forms a cohesive ball and is smooth and elastic.
Turn the dough out of the mixer and onto the countertop. Cut the dough into twelve equal pieces.
Shape each piece of dough into a ball.
Place the buns on a parchment-lined baking sheet. Lightly push down to form a wider burger shape.
Cover the rolls and place them in a warm place to rise for 30 minutes.
Preheat the oven to 425ºF.
Mix together the egg wash: crack an egg and add a splash of water. Whisk together.
Lightly brush the egg wash over each bun. Sprinkle with sesame seeds if desired.
Bake the buns for about 15-18 minutes until golden brown and reach 190ºF inside.
Allow to cool completely before slicing and enjoying with your burgers.
