Place the warm water in the bowl of a stand mixer. Sprinkle the yeast and sugar over the warm water, stir gently with a whisk and let sit for 5 minutes or until the mixture looks frothy.
Melt the butter. Add the flour, salt and melted butter to the mixer bowl.
Using the dough hook attachment on your stand mixer, mix on low speed until the ingredients come together. Increase the speed on the mixer to medium and mix for about 5 minutes or until the dough pulls away from the sides of the bowl and looks smooth.
Turn the mixer off and remove the dough from the mixer bowl. Place the dough in an oiled bowl and cover with plastic wrap. Let rise for an hour.
Remove the plastic wrap from the bowl. Punch down the dough.
Preheat the oven to 450 degrees F.
Place the dough on your work surface. Divide the dough into 8-16 pieces depending on how big or small you want the pretzels to be.
Place parchment paper on two baking sheets and lightly oil it.
Set a large, wide pan (like a dutch oven or 12 inch saute pan) on the stove over medium-high heat and add 10 cups of water and the baking soda. Let the water come to a boil while you're shaping the pretzels.
Roll out the dough into a long (8-12 inch) rope. Then form into a pretzel shape: form the rope into a heart with the ends of the dough touching the bottom of the "heart". Twist the ends of the dough together and pinch them to the bottom of the pretzel. Place on the parchment lined ½ sheet pan.
Continue forming the pretzels until you've used all the dough.
Place the pretzels gently and carefully, one-by-one into the pan of baking soda and boiling water. After 30 seconds, remove them from the water with a flat spatula that has slots in it (allows the water to drain off as you lift the pretzel from the water). Place them back on the parchment lined ½ sheet pan.
Next brush them with the egg wash. Then sprinkle with coarse sea salt.
Place the pretzels in the preheated oven and bake until they're a dark, golden brown, about 10-15 minutes depending on the size of the pretzels you made.
Once they're done, remove them from the oven and allow them to cool for 5-10 minutes.
Notes
We divided the dough into 12 pieces. If you would like smaller pretzels I'd recommend dividing the dough into 16 pieces.