In a small saucepan, heat the water and butter until butter is melted. Remove from heat and let cool for 5 minutes, or until 110-120 degrees F.
In the bowl of stand mixer fitted with a dough hook attachment, combine 2 cups of the all-purpose flour, yeast, salt, sugar, rosemary, and cheese. Add the water mixture and mix until combined to form a wet dough.
With the mixer on medium low, add the remaining flour 1/4 cup at a time until dough clears the bowl but is still slightly sticky to the touch. You may not need to use all of the flour. Continue to knead until dough is smooth and elastic, about 5-7 minutes.
Place dough in a grease bowl and cover with plastic wrap. Let rise in a warm place until doubled, about 1 hour.
Gently punch down dough and knead a few times. Tear a small handful of dough off with your hand and roll into a long thin rope about 1/2 inch wide. Use a sharp knife to cut into 1-inch pieces. Repeat with remaining dough.
Preheat oven to 450 degrees and line a sheet pan with parchment paper.
Line another sheet pan with parchment paper and liberally spray with non-stick cooking spray. Pour the 10 cups of water into a large pot and set over medium high heat. Stir in the baking soda. Bring the mixture to a boil and then cook the dough pieces in small batches for 30 seconds. Remove with a slotted spoon and place on the greased sheet pan.
Place par-boiled dough pieces on the clean parchment paper-lined sheet pan, and brush with egg wash and sprinkle generously with coarse salt. Bake until golden brown, about 10-12 minutes. Let cool completely.
To make the dipping sauce, whisk all ingredients together in a small bowl and serve with the pretzel bites. Pretzel bites are best the day they are made but will keep for a few days in an airtight container at room temperature.