In a medium bowl, mix together warm water, yeast, and 1/2 cup brown sugar; let stand until foamy, 5 to 10 minutes.

In the bowl of an electric mixer, combine flour and coarse salt using your hands.

Add butter and continue to combine with your hands until mixture is crumbly.

Add yeast mixture and, still using your hands, combine until a shaggy dough is formed and water is absorbed.

Using the dough-hook attachment, mix dough on medium-low speed until tight, elastic, and smooth, 6 to 8 minutes.

Wrap in plastic and refrigerate at least 8 hours and up to overnight.

Preheat oven to 450 degrees, with rack in upper third.

Lightly coat a baking sheet with cooking spray.

Roll out dough into a 14-by-12-inch rectangle. Cut dough into twelve 14-inch-long strips, each about 1 inch wide. Working with one piece at a time, form dough into pretzel shape.

Transfer to prepared baking sheet.

Coat a second large baking sheet with cooking spray.

In a wide stockpot, combine 8 cups water, baking soda, beer, and remaining 1/4 cup brown sugar; bring to a simmer over medium-high.

Simmer pretzels, one at a time (or two, with smaller pretzels), about 30 seconds each, holding them under surface of water, use a slotted spoon or spatula.

Transfer to prepared baking sheet. (You can gently reshape pretzels if they become misshapen.)

Sprinkle pretzels with pretzel salt.

Transfer to oven and bake 5 minutes.

Rotate baking sheet and bake until deep brown, 3 to 6 minutes more. Transfer pretzels to a wire rack; let cool slightly.

Serve warm with mustard.