Stir together the water and sugar in a large bowl. Sprinkle the yeast evenly over the top and let stand for about 5 minutes, until foamy.
Stir egg, butter and salt into the proofed yeast. Then stir in the 4 ¼ cups of the flour, 1 cup at a time, continuing to stir until the dough comes together into a rough mass.
Lightly flour a work surface and turn the dough out onto it. Knead the dough, adding more flour as needed to prevent dough from sticking to the surface, for 8 to 10 minutes, until the dough is smooth, soft, and elastic.
Shape the dough into a ball and place into a lightly greased mixing bowl. Cover with a clean, damp towel and place in a warm area until dough doubled in size, about 1 hour.
Preheat oven to 400˚F.Line 2 baking sheets with parchment paper.
Punch the dough to release the air and cut the dough into 12 equal pieces. Working with 1 piece at a time, roll the dough into a rope 12-16 inches long, then twirl each pretzel into shape, twisting the final ends a few times before sealing.
Combine water and baking soda in a deep stockpot and bring to a boil over high heat. One at a time, dip pretzel-dogs into boiling water for about 30 seconds, then carefully return them to baking sheets.
In a small bowl combine beaten egg and 1 tablespoon water.
Generously brush pretzels with egg wash and sprinkle with furikake seasoning and Parmesan. Bake the pretzel-dogs for 18 to 22 minutes or until they are golden brown.
Transfer the pretzels to a wire rack and cool for a few minutes before serving.
Queso Dip: Melt butter in a small pot over medium heat. Add flour and whisk together for 2 minutes. Stir in milk and continue to whisk for 2 to 3 minutes or until mixture is smooth. Remove from heat and whisk in shredded cheese, small handfuls at a time, until fully incorporated and sauce is smooth. Season queso dip with salt and pepper and serve immediately with warm pretzels.