Combine warm water with brown sugar, stir until sugar is almost dissolved. Add yeast, stirring until incorporated.
Set aside in a warm place for 10 minutes, or until foamy.
In a large bowl, combine flours, protein powder, ¼ cup potato starch, xanthan gum, baking powder, and salt.
Add oil and egg, stirring until well distributed. Pour in yeast mixture, stirring until well combined - it will be VERY wet.
Generously spray a large, clean metal or glass bowl with nonstick spray.
Add dough to the bowl, loosely cover with plastic wrap, set aside in a warm spot to rise for an hour or two, until doubled in size.
Once the dough has doubled in size, stir remaining potato starch into the dough.
Using lightly greased hands, stir/knead until no longer *super* sticky - it'll be a little sticky. Use a little more potato starch if needed.
Return to bowl, loosely cover with plastic wrap, and allow to rise for another 30 minutes.
Divide dough into 4 equal portions.
Lightly dust a clean surface with potato starch, roll out dough ball into a longish "snake".
Twist into a pretzel shape, pinch cross points to secure. Be gentle when working with the dough - the roll will break easier than full gluten dough does.
Allow pretzels to rise for another 30 minutes in a warm place.
Preheat oven to 375 F (190 C). Line a baking sheet with parchment paper, set aside.
Bring water to a gentle boil in a large pot. Add baking soda, stir until combined.
Gently transfer the pretzels - one or two at a time - to the pot of water; cook for about a minute on each side.
Gently stir as they boil, to ensure that the entire surface of each is in contact with the water at some point.
Use a slotted spoon to transfer each pretzel to prepared baking sheet.
Whisk together egg and water, brush this mixture across the tops of the pretzels.
Generously sprinkle the tops with coarse sea salt.
Bake for about 25 minutes, until golden brown.
Allow to cool for a minute or two before eating, serve with Mustard-Cheddar Dip