Add the yeast to 2 cups warm water (no hotter than 110 degrees F) in a bowl and whisk until dissolved. Slice 1/2 stick (4 tablespoons) of the butter into thin slices. Lightly whisk the butter slices and brown sugar into the yeast mixture, then fold in 2 cups of the flour with a rubber spatula until combined. Add the salt and remaining 2 1/2 cups flour and mix until combined; the dough will be sticky.
Place the dough on a floured surfaced and knead until smooth but still slightly sticky, 8 to 10 minutes. Place in a bowl that has been sprayed with cooking spray and cover with plastic wrap. Allow the dough to double in size, about 1 hour 30 minutes.
Preheat the oven to 425 degrees F. Grease a baking sheet with cooking spray.
Deflate the dough by punching it in the center, then, on a lightly floured surface, divide it into 12 pieces. Roll the pieces until they are about 1/2 inch thick, then form each into a pretzel shape.
Whisk the baking soda into 4 cups warm water until dissolved. Dip each pretzel into the mixture, then place on the prepared baking sheet. Sprinkle with pretzel salt and bake until golden brown, about 10 minutes.
Meanwhile, melt the remaining 1 1/2 sticks butter and transfer to a shallow dish. When the pretzels are done, dip the salt side of each in the butter. Place on a drying rack and allow to cool slightly. Serve warm.