Make the pretzel dough: In a large mixing bowl, whisk together the bread flour, brown sugar, and salt. Sprinkle the instant yeast over the dry ingredients and whisk again to distribute it evenly. Pour in the warm water and melted butter.
Bring the dough together: Use a sturdy spoon or spatula to stir until a shaggy dough forms and most of the flour is moistened. If the dough looks very dry and there are still pockets of loose flour at the bottom, drizzle in another 1–2 tbsp of warm water, a little at a time.
Knead until smooth: Turn the dough out onto a lightly floured counter. Knead for about 6–8 minutes, adding just enough flour to keep it from sticking, until smooth and slightly tacky.
First rise: Shape the dough into a ball and place it in a lightly oiled bowl, turning once to coat. Cover and let it rise 45–60 minutes (up to about 75 minutes in a cool kitchen), or until roughly doubled.
Prep pans and oven: Line two baking sheets with parchment paper and lightly grease the parchment. Preheat the oven to 220°C / 425°F with two racks near the center.
Divide the dough: Gently punch down the dough, turn it out, and cut into 4 equal pieces. Keep pieces you’re not shaping loosely covered so they don’t dry out.
Shape ropes: Roll each piece into a rope about 45–50 cm (18–20 inches) long, resting the dough 3–4 minutes if it snaps back.
Form mini pretzels or bites: Cut each rope into 8–10 pieces for mini knots (roll and twist) or into 2–3 cm (1-inch) nuggets for bites. Arrange on the prepared trays with a little space between pieces.
Rest briefly: Cover trays loosely and rest 10–15 minutes while you prepare the baking soda bath.
Set up the baking soda bath: Bring 1 litre (4 cups) water to a gentle boil in a wide, deep saucepan. Carefully stir in the baking soda (it will foam). Reduce heat to a steady simmer.
Boil the pretzel pieces: Working in batches, simmer pretzels 20–25 seconds, turning once. Lift with a slotted spoon, drain well, and return to the baking sheets.
Egg wash and salt: Brush lightly with egg wash and sprinkle with coarse salt to taste.
Bake: Bake 10–14 minutes, rotating pans halfway through, until deep golden brown.
Cool slightly: Transfer pretzels to a wire rack and cool at least 5 minutes before serving.
Start the cheese dip base: In a medium saucepan over medium heat, melt the butter. Whisk in the flour and cook 1–2 minutes until smooth and slightly foamy (not browned).
Add milk gradually: Pour in the milk in 3–4 additions, whisking smooth each time. Cook and whisk 3–5 minutes, until thickened to the consistency of light cream.
Melt the cheeses: Reduce heat to low. Whisk in cream cheese until melted, then add cheddar in small handfuls, whisking until fully melted and smooth.
Season the dip: Stir in Dijon mustard, smoked paprika, salt, and black pepper. Adjust to taste; thin with a splash of warm milk if needed.
Serve warm: Transfer dip to a heatproof bowl and serve immediately with warm mini soft pretzels. Keep dip over very low heat if needed, stirring occasionally.