Combine water, sugar, kosher salt, and yeast in a stand mixer, let sit for 5 minutes for yeast to proof. When it begins to foam, continue.
Use a hook attachment, and turn your mixer on. Add flour in a little at a time, with melted butter.
Mix on low to combine ingredients.
Add in chopped sun dried tomatoes.
Increase to medium speed, and knead until dough pulls away from the sides of the bowl (5 minutes).
Remove from bowl, put a little oil in bowl, and put dough back in, cover, and let rise until double in size.
Preheat oven to 450 degrees F
Line sheet with parchment paper, and spray with cooking spray, or use a silicone baking mat
Add 9 cups of water and baking soda to a large pot, and put on high heat and bring to a boil.
While waiting for soda bath to boil, divide your dough into 12 even balls, and form into pretzel twists with half a cheese stick in the center. Make sure it is completely covered. (I split the stick down the middle, flattened my dough ball, and elongated it, then stuck the cheese downt he center, and rolled it into a log around the cheese, then twisted the pretzel into shape)
One at a time, place each twist into boiling water for 30 seconds.
Remove, and put on cookie sheet, about an inch or so apart.
Beat egg yolk and water together and brush over top of pretzels
Sprinkle with salt and bake 11-15 minutes until golden brown.
Allow to cool for 5-10 minutes, then serve.