In the bowl of a stand mixer add warm water and instant yeast. Whisk together.
Add the butter, brown sugar, and salt to the yeast mixture. Whisk until combined.
Fit the dough hook on the mixer and turn on low speed. Slowly add in 3 cups of flour, 1/2 cup at a time, until mixture is combined and pulling away from the sides. PRO TIP: If dough is still sticky, add flour, and an additional 1/4 cup at a time until no longer sticky and dough bounces back when poked with your finger.
Place the dough out onto a lightly floured surface. Knead by hand for 2 minutes and then shape into a ball. Place in a large mixing bowl that has been lightly coated with cooking spray. Cover and let rest for 10 minutes.
While your dough is resting, bring a large pot of water and baking soda to a boil. Cut dough into six equal pieces. Roll each piece into 1 inch long ropes. Cut each rope into 1 1/2 inch bites.
Preheat the oven to 425 degrees and line two baking sheets with parchment paper.
Add each piece of cut dough into the boiling water (work in batches) and let cook for 20 seconds. Remove from the water with a slotted spoon and place pretzel bites on the baking sheets evenly spaced about an inch apart.
In a small bowl combine egg and water. Beat together. Brush beaten egg wash over each pretzel bite and then sprinkle with coarse salt.
Place the baking sheet in the oven and let bake for 12 to 15 minutes, or until golden brown.
In a medium saucepan over medium-low heat the butter. Once the butter is melted add flour and whisk together until no lumps remain.
Turn heat to medium and slowly whisk in milk. Let cook over medium heat whisking often for about 5 minutes. The mixture will thicken as it cooks.
Add cheese, salt, pepper, and ground mustard. Whisk to combine until smooth. Season with additional salt and pepper to taste.