Add melted butter, salt, and 4 cups flour. Mix to a shaggy dough, then knead 6–8 minutes until smooth and elastic, adding flour a tablespoon at a time only if sticky.

Cover and let rise in a warm spot 45–60 minutes, until doubled.

Divide dough into 6 pieces. Roll each into a 16-inch rope and cut into ~1-inch pieces (about 48 bites). Heat oven to 425°F and line two baking sheets with parchment.

Bring the 10 cups water and baking soda to a gentle boil. Boil bites 8–10 at a time for 20–30 seconds, then lift onto the parchment with a slotted spoon.

Brush each bite with egg wash and sprinkle with coarse salt. Bake 12–15 minutes, until deep golden brown. Cool slightly.

Meanwhile, melt 2 tbsp butter, whisk in the flour, and cook 1 minute. Slowly whisk in the warm milk and simmer 2–3 minutes until thickened.

Off the heat, stir in the cheddar a handful at a time until smooth. Season with salt, mustard powder, and paprika. Serve the pretzel bites warm with the cheese sauce.