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 the egg, butter and salt into the proofed yeast. Then stir in the 4 ¼ cups of the 1 cup at a time, continuing to stir until the dough comes together in 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 10 equal pieces. Working with 1 piece at a time, roll the dough into a rope 12-16 inches long, then wrap the rope around a hot dog, leaving a 1/2 inch on each end exposed, and pinch the dough on each end to seal.
Combine the water and baking soda in a deep stockpot and bring to a boil over high heat. One at a time, dip the pretzel-dogs into the boiling water for about 30 seconds, then carefully return them to the baking sheets.
Generously brush the pretzels with the egg wash and sprinkle with the black sesame seeds, sunflower seeds and sea salt. Bake the pretzel-dogs for 16 to 18 minutes, until they are golden brown. Transfer the pretzels to a wire rack and let cool for at least 10 minutes before serving.
Cheese sauce: 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. Add mustard and cheese and whisk together until smooth. Season with salt and pepper and serve immediately with pretzel dogs.