Place water, brown sugar, salt, and yeast together in a bowl of a stand mixer fitted with a hook attachment. Let it sit for 5 minutes or until it begins to foam.
Add the flour and butter and mix on low speed until combined. Turn up speed and beat on medium for about 5 minutes or until dough pulls off the sides and is no longer sticky.
Remove dough from bowl, grease bowl lightly with oil, and place dough back in the bowl and cover with a tea towel. Let rise for 50 to 60 minutes or until double in bulk.
Preheat oven to 450℉. Bring 10 cups of water in a large pot to a boil. Line 2 baking sheets with parchment paper and lightly grease.
Lightly flour working surface. Divide dough into 8 pieces and roll each into 24 inch ropes and then shape into pretzels. Place on parchment lined pan.
Add the baking soda to the boiling water and cook each pretzel one at a time in the boiling water for 30 seconds. Remove with a slotted spatula and place back on the baking pan.
Mix the egg yolk and water together in a small dish and brush over the top of pretzels. Sprinkle with coarse salt and then bake (one pan at a time) for 13 to 15 minutes until dark golden brown. Remove from oven and let cool on a rack for 5 minutes before serving.