In the bowl of a stand mixer, combine yeast, 1 teaspoon of the brown sugar and ¼ cup water. Let this sit for about 5 minutes, until thick and frothy.
Add the remaining brown sugar, water and salt. Stir with a wooden spoon to combine. Add 3 cups of the all-purpose flour. Using the dough hook attachment, start to mix the dough using low speed, then increasing gradually until all the flour has been mixed in.
Add 1 cup of flour to the mixing bowl. Knead until the dough starts to clean the sides of the bowl and making slapping sounds, about 6-9 minutes.
If the dough is still sticky, add more flour gradually in small amounts. The dough is ready when it cleans the sides of the bowl, and you can stretch it thinly without breaking right away (aka the windowpane test). Do not add too much flour. The should be soft, elastic and smooth, not dry.
Allow the dough to rest for 10 minutes. Preheat your oven to 400 ℉. Line 2 large baking pans with parchment paper.
Turn the dough out into a lightly floured surface. Cut the dough into 12 equal portions. To shape the pretzels, roll each portion into a log that is 22 inches long.
Next, make a U shape with the log. Take the two ends of the U and put one end over the other. Twist the ends one time. Holding the two ends of the twist, fold the dough downwards so that the tip of the rope meets the base of the letter U. Pinch the ends to seal. Arrange the shaped pretzels, shaped evenly on two baking pans. I put 6 in each.
In a large pot, combine 10 cups of water and ½ cup baking soda. Bring to a boil. Boil the pretzels, a few at a time, in the pot for 15 seconds. Drain as much liquid as you can then put them back in the pan.
Beat together egg and 1 tbsp of water. Brush the pretzels with this egg/water mixture. Sprinkle coarse salt all over the boiled pretzels.
Bake at 400 F for 12-15 minutes, or until he exterior is nicely golden. Brush with melted butter as soon as they are out of the oven.