In a small bowl, add the warm water and yeast. Stir to dissolve. Cover and let proof (get foamy and bubbly) for about five minutes. If using instant yeast you can still dissolve it in the water but it's not necessary.
In the bowl of a stand mixer with the dough attachment, add the brown sugar, butter, salt, and flour. Stir together. Pour in the yeast/water mixture and mix until dough starts to come together. Keep kneading the dough for about 3 minutes adding up to 1/2 cup of flour as needed (you most likely will not need much) until the dough comes away from the sides and is not sticky.
Spray a large bowl with cooking spray and place the ball of dough in it. Cover lightly with a dish towel and let rise about 30-45 minutes. It should be puffy but should not be doubled in size (rising too much will lead to a more bread like pretzel).
In a pot large enough to hold the size of your pretzels, bring 8 cups of water to a boil with the baking soda. While water is comes to a boil, form the pretzels.
On a lightly floured surface, place the dough and cut into 10-12 even pieces. Roll each piece out one at a time into about a 22-inch long ropes. Form into a pretzel shape by crossing the ends over twice or cut rope into 1½ inch sections for pretzel bites. See pictures above.
Line two baking sheets with parchment paper. Gently lower each pretzel one at a time into the pot for 15-30 seconds. Remove with a slotted spoon, letting the water drain then place on the parchment lined baking sheets. Repeat with all of the pretzels. Stir together the egg and water. Brush the pretzels with the egg wash and sprinkle with coarse salt. Cover and let pretzels rise 15 minutes. Preheat oven to 450℉ while pretzels rise.
Bake pretzels for 8-10 minutes or until pretzels are a deep golden brown. Remove from the oven and brush with the 2 tablespoons of melted butter generously. Eat plain or with a cheese sauce. See notes for cheese sauce recipe. For cinnamon sugar option, brush with butter and sprinkle with cinnamon sugar.
Notes
Cheese Sauce:
1 Tablespoon butter
1 Tablespoon flour
3/4 cup milk
1 cup shredded cheddar cheese
1/2 teaspoon garlic powder
salt to taste
Melt butter, whisk in flour (cook 1 minute).
Slowly whisk in milk until smooth and slightly thick.