Place 1 cup water, butter, sugar, salt, flour and dry yeast in bread machine pan in order listed. Do not mix.
Select Dough setting and press start.
When cycle is complete, remove dough and transfer to lightly floured surface.
Preheat oven to 400º F. Line baking sheet with parchment paper and lightly spray with cooking spray.
Divide dough into 6 equal portions.
Roll each piece of dough into a 18-20 inch rope.
Shape rope into U, take 2 ends and cross over 1 or 2 times. Bring end towards you, fold down leaving ends over-hang a bit. Press down a bit where dough intersects at the bottom.
Prepare baking soda water. Using large pot bring 6 cups of water to rapid simmer. Add baking soda slowly to water, stir to dissolve. Reduce heat to maintain a simmer.
Place 1 to 2 pretzels into water. Do not overcrowd pot.
Cook for 30 seconds, flip and cook for another 30 seconds.
Using large slotted ladle remove pretzels and transfer to baking sheet.
Prepare egg wash by whisking together egg with 1 tsp of water.
Brush pretzels with egg wash. Sprinkle with coarse salt. My favorite is coarse sea salt.
Bake 400º F for 10-12 minutes until golden brown.
Remove from oven and brush generously with melted butter.