To a bowl of a stand mixer fitted with a paddle attachment, add the warm milk, room temperature sourdough discard, egg, egg yolk, melted butter and sugar. Mix on low speed until smooth.
To the wet ingredients, add the flour, instant yeast and salt. With the hook attachment, mix again on low speed. Stop to scrape the sides of the bowl to incorporate all the dry ingredients and then continue to mix on low for 1-2 minutes.
Continue to knead the dough on medium speed for 6-8 minutes. You’ll know the dough has been kneaded completely when it is smooth and elastic.
Remove the dough and place it in a lightly greased bowl and cover.
Place the dough in a warm place in your kitchen and allow the dough to double in size.
Once the dough has at least doubled in size, punch the top of the dough to remove all the air bubbles. Then, place the dough on a lightly floured work surface.
Gently stretch the dough into a rough rectangle shape. With a bench scraper, divide the dough into 10 equal pieces (about 100g each).
Pull the corners of the dough towards the center, flip, and then roll each piece of dough into a round, tight ball. Place them on a rimmed baking sheet lined with parchment paper. Slightly flatten them.
Once shaped, cover the baking sheet with this plastic cover and let the dough rest for its second rise.
Preheat the oven to 375 degrees F. In a small bowl, whisk together an egg and a splash of water, then brush the top of each bun with the egg wash.
Bake for 13-15 minutes or until the buns are beautifully golden brown and the internal temperature is 190 - 200 degrees F.
Remove the Sourdough Buns from the oven and let them cool on a cooling rack. Serve the Sourdough Burger Buns at room temperature with your favorite grilled meat and toppings of choice.
