Dissolve the yeast and sugar in one cup of warm water. Let sit for five minutes, or until it is frothy on top. Meanwhile, stir together two cups of the flour and 1/2 tsp of salt in a large bowl.
Once the yeast water is frothy, pour it into the bowl with the flour and salt. Stir the ingredients together until they form a shaggy ball of dough. Turn the dough out onto a floured surface and knead for 8 minutes, incorporating an additional 1/2 to 1 cup of flour as you knead.
After the dough has been kneaded for 8 minutes, it should be smooth, elastic, and will spring back when poked with a finger. Place the dough in a greased bowl, cover lightly, and let rise for one hour or until double in size.
Once the dough is double in size, punch the dough down, turn it out onto a clean surface, and flatten it into a disc. Cut the disc into 16 equal sized wedges. One by one, roll each wedge into a long rope, about 1/2 in diameter (about 18 inches long). Cut the rope into one inch sections, and place them on a baking sheet that has been covered with parchment and lightly misted with non-stick spray (you'll need two sheets to fit all the pretzel bites).
Let the pretzel bites rise for 30 minutes, or until soft and pillowy. Preheat the oven to 425 degrees. Bring a large pot of water to a boil over high heat. Once boiling, carefully addd 1/4 cup of baking soda (it will fizz and bubble furiously for a moment, so be careful).
Add 15-20 of the pretzel bites at a time to the boiling baking soda water and boil for about 90 seconds, or until puffed and a few begin to crack. Stir lightly as they boil to make sure all sides of the pretzel bites contact the boiling water. Remove the boiled pretzel bites with a slotted spoon and let drain on a wire cooling rack. Repeat in small batches until all of the pretzel bites are boiled.
After the bites have drained for a few minutes, return them to the prepared baking sheets. Whisk together one egg with one tablespoon of water. Use a pastry brush to liberally brush the egg mixture over the boiled pretzel bites. If you want to make plain pretzel bites, sprinkle with coarse sea salt after brushing with the egg wash.
Bake the pretzel bites for 15-17 minutes, or until deeply golden brown and shiny. Allow the pretzel bites to cool for a few minutes.
In a small bowl, stir together the Parmesan, garlic powder, salt, and some freshly cracked pepper. In a separate small bowl, melt the butter in the microwave (microwave for 20 seconds, then stir until completely melted).
Place the baked pretzel bites in a large bowl and pour the melted butter over top. Toss to coat. Add the parmesan topping and toss to coat again. Serve immediately.