Combine the warm water, brown sugar, and kosher salt in the bowl of a stand mixer. Sprinkle the yeast on top. Let sit for 5 minutes until it begins to foam.

Add the melted butter and flour. Using the dough hook attachment, mix on low speed until combined. Increase speed to medium and knead until the dough is smooth and pulls away from the sides of the bowl.

Remove the dough, shape it into a ball, and place it in a lightly oiled bowl. Cover with a clean towel or plastic wrap and let sit in a warm place until doubled in size.

Preheat the oven to 450°F (230°C). Line two baking sheets with parchment paper and lightly spray with non-stick spray or brush with oil.

Bring the 10 cups of water and the baking soda to a rolling boil in a large pot.

Turn the dough out onto a lightly oiled work surface and divide into 8 equal pieces. Roll each piece into a 24-inch rope. Make a U-shape, cross the ends over each other twice, and press them onto the bottom of the U to form the shape of a pretzel.

Place the pretzels into the boiling baking soda water, 1 or 2 at a time, for 30 seconds. Remove with a slotted spatula, let excess water drip off, and place onto the prepared baking sheets.

Brush the top of each pretzel with the egg yolk and water mixture, then generously sprinkle with the coarse pretzel salt.

Bake in the preheated oven until dark golden brown.

While the pretzels bake, make the beer cheese. Melt the butter in a medium saucepan over medium heat. Whisk in the flour and cook for 1 minute.

Gradually whisk in the milk and beer, ensuring there are no lumps. Stir in the mustard and garlic powder. Simmer until thickened.

Remove from heat. Add the grated cheddar cheese a handful at a time, stirring constantly until completely melted and smooth. Serve warm with the freshly baked pretzels.