In a standing mixer bowl, add the warm water, yeast, and brown sugar, and let sit for about 5 minutes (it should be foamy*). Then add in the melted butter and salt. Mix with a rubber spatula by hand. Gradually add in the flour, mixing with a rubber spatula until it becomes dough and is no longer sticky.
Add the dough hook to the standing mixer, and mix the dough on medium-low speed for 4-5 minutes, until the dough is smooth and pulling away from the bowl.
Spray a large bowl with cooking spray, add the dough, and cover with a towel. Place the bowl in a warm place, and let it rise until it's double in size (about 55 minutes to an hour).
Preheat the oven to 400° F and line two baking sheets with parchment paper. Set aside.
Once the dough is done rising, cut it into 12 even sections. Roll each section of dough into a 20 inch long rope. Form a U-shape with the rope, then grab hold of the top ends. Cross them over each other twice. Bring the two ends down over the bottom of the U- shape, pressing the ends down to form the pretzel. (Linked above is a tutorial that shows how to do this).
Add the water and baking soda to a large pot and bring to a rolling boil. Using a large spatula (or slotted spoon), carefully place the pretzels (one at a time) into the boiling water for 20 seconds, then remove. Don't leave them in for too long! Drip off as much excess water as possible, and then place the pretzels on the parchment paper-lined baking sheets.
Whisk together the egg and 1 tablespoon of water, forming an egg wash. Brush the egg wash over the top of each pretzel using a pastry brush. Then generously sprinkle coarse sea salt over each pretzel.
Bake one baking sheet of pretzels at a time, for about 12-15 minutes until golden. Immediately after baking, butter the top of each pretzel with unsalted butter.
Allow to cool for a few minutes, and then serve while they are still warm with some cheese dip (below). Enjoy!