In the bowl of a stand mixer stir together water, yeast, brown sugar and butter until combined. Let rest for 5 minutes or until small bubbles have formed.
Add flour and salt to the yeast mixture, and using the dough hook attachment mix on low speed until the ingredients are combined. Increase to medium speed and mix for 2-3 minutes, dough should be smooth and separated from the bowl.
Remove dough from bowl and knead into a ball. Add vegetable oil to a large bowl and add the dough, tossing to ensure dough is coated. Cover and let rise for 1 hour or until doubled in size.
Preheat oven to 425 degrees. In a large pot bring water to a boil and add baking soda.
Divide the dough into 8 sections, roll each section into a rope about 20 inches long. Cut each rope into 1 inch bites.
Working in batches, boil the pretzel bites for 30 seconds, remove with a mesh strainer or slotted spoon and place on a baking sheet. Create an egg wash by mixing the egg and 1 tbsp of water and brush the top of each pretzel bite, sprinkle with coarse sea salt.
Bake for 16 - 20 minutes until golden brown brown. Let cool for 5 minutes before serving.