In a large mixing bowl, combine the warm water, yeast, and sugar, stir briefly, and let sit for 5–10 minutes until foamy.

Stir in the melted butter and fine salt, then add 2 1/2 cups of the flour and mix until a soft dough forms.

Add the remaining 1/2 cup flour a little at a time, mixing until the dough pulls away from the sides of the bowl but is still slightly tacky.

Turn the dough out onto a lightly floured surface and knead for 5–7 minutes until smooth and elastic.

Place the dough in a lightly greased bowl, cover, and let rise in a warm place for about 1 hour, or until doubled in size.

While the dough rises, pat the hot dogs dry with paper towels and cut the cheddar cheese into 8 thin sticks.

In a large pot, bring 8 cups of water to a gentle boil, then carefully stir in the baking soda until dissolved; reduce to a simmer.

Punch down the risen dough and turn it onto a lightly floured surface, then divide it into 8 equal pieces.

Roll each piece of dough into a 14–16 inch rope, then flatten each rope slightly with your fingers.

Place a cheese stick alongside each hot dog, then wrap a dough rope snugly around each hot dog and cheese, pinching seams and ends to seal.

Working in batches, carefully lower each wrapped dog into the simmering baking soda bath for about 30 seconds, then remove with a slotted spoon and place on a parchment-lined baking sheet.

In a small bowl, whisk together the egg and 1 tablespoon water to make an egg wash.

Brush each pretzel dog lightly with the egg wash and sprinkle with coarse salt.

Bake in a preheated 425°F oven for 12–15 minutes, or until deep golden brown and the cheese is just starting to bubble at the seams.

Let cool for 5–10 minutes before serving to allow the pretzel exterior to set and the cheese to firm slightly.