Combine _ cup of warm water and the yeast in the bowl of an electric mixer fitted with the dough hook, mixing until the yeast is dissolved. Add the sugar, salt, and flours and mix on low speed for 1 to 2 minutes, until the dough comes together. Increase the mixer speed to medium and beat for another 4 to 5 minutes, until the dough forms a smooth ball. Transfer the dough to a lightly greased bowl, cover with a clean dish towel, and let rise in a warm, draft-free area for about 1 hour, until slightly puffed up.
Line two baking sheets with parchment paper. Stir together the baking soda and remaining cup of warm water. Divide the dough into 12 pieces and roll each into a ½-inch thick rope. Cut each rope into about four 1-inch pieces. Dip each piece of dough into the baking soda water and place on the prepared baking sheets. Cover with the dish towel and let rise for 30 minutes.
Preheat the oven to 450°F. Bake the pretzels for 8 to 10 minutes, until golden brown. Remove from the oven and, while the pretzels are still warm, brush the tops with melted butter. Sprinkle each pretzel bite with salt.
Melt the butter in a medium saucepan over medium heat. Stir in the flour and cook for 2 minutes. Slowly add the milk and beer, whisking constantly until smooth. Cook for 4 to 6 minutes, stirring constantly, until the mixture thickens enough to coat the back of a wooden spoon. Remove from the heat and stir in the cheddar cheese, mustard, pepper, and nutmeg.
Serve the pretzels with the cheddar sauce alongside for dipping.