Combine warm water, yeast, and sugar in a bowl. Let sit 5 minutes until foamy. Add gluten free flour, xanthan gum, salt, and melted butter. Mix until a soft dough forms, then knead gently on a floured surface for 2 minutes. Cover and let rise in a warm spot for 45 minutes.
Preheat oven to 425F and line a baking sheet with parchment paper. Divide dough into 6 equal pieces. Roll each piece into a 16-inch rope and twist into a pretzel shape, pressing the ends firmly to seal.
Bring 6 cups of water and baking soda to a rolling boil. Carefully lower each pretzel into the boiling water for 30 seconds per side, then transfer to the prepared baking sheet.
Brush each pretzel with beaten egg and sprinkle generously with coarse sea salt.
Bake for 14 to 16 minutes until deep golden brown. Remove from oven and immediately brush with melted butter. Serve warm with mustard or cheese dip.