Whisk together the warm water, sugar, and yeast in a large mixing bowl. Cover the yeast mixture and allow it to sit for 5 minutes or until foamy on top.
As the yeast activates, combine the melted butter, 4 cups of all-purpose flour, and salt into the bowl of a large stand mixer fitted with the dough hook attachment.
Once the yeast mixture is foamy and activated, add it to the flour mixture.
Knead the dough at low speed for about 1 minute, scraping down the sides of the bowl with a rubber spatula if needed. Continue to beat at low speed until the dough comes together and pulls away from the bowl (Note: if the dough is extremely sticky, add 1/4 cup of additional flour at a time). Continue to beat at medium speed for an additional 5 minutes until the dough is smooth. Poke the dough with your finger – it should slowly bounce back.
Transfer the dough to a lightly oiled bowl, turning to coat it with the oil. Cover the bowl with a damp towel or plastic wrap and place it somewhere warm. Let the dough rise for 30-60 minutes or until it has doubled in size.
Preheat the oven to 450°F. Line two large baking sheets with parchment paper and set aside.
Bring 3 quarts of water to a boil. Carefully add the baking soda (it will boil over, so add it slowly).
To make the pretzel bites, turn the dough out onto a countertop or clean work surface that has been very lightly dusted with flour. Divide the dough into eight equal pieces. Roll each piece into a long 10-12-inch rope about 3/4-inch thick. Slice each rope into small 1-1.5-inch long pieces. Repeat with the remaining dough.
Use a slotted spoon to carefully lower 10-12 dough bites into the baking soda bath and let them boil for 15-30 seconds. Remove the boiled pieces, allowing as much water to drip off as possible, before transferring them to the prepared baking sheet. Repeat with another batch of pretzel bites.
In a small bowl, whisk together one egg with two tablespoons of water. Brush the tops of each pretzel bite with the egg wash, then sprinkle with coarse sea salt.
Bake the first batch of pretzel bites for 14-18 minutes or until the tops are golden. Meanwhile, repeat the boiling/egg-washing/salting with the remaining dough pieces and bake once the first pan is finished.
For extra buttery pretzel bites, brush the tops with melted butter while they are still warm and just before serving.