In a stand mixer, combine warm beer, sugar, and yeast. Let sit 5-10 minutes until foamy.
Add bread flour, salt, and melted butter. Mix on low until a shaggy dough forms, then knead on medium 5-7 minutes until smooth and elastic.
Transfer dough to a lightly oiled bowl, cover, and let rise 1 hour until doubled.
Preheat oven to 425°F (220°C) and line baking sheets with parchment paper.
Punch down dough, divide into 12 pieces, roll each into a 20-24 inch rope, and shape into pretzels.
Bring 10 cups water to a boil, add baking soda carefully. Boil each pretzel 30 seconds per side. Remove with slotted spoon to baking sheets.
Sprinkle pretzels with coarse sea salt and bake 12-15 minutes until golden brown.
For the dip, melt butter in a medium saucepan over medium heat. Whisk in flour and cook 1 minute.
Slowly whisk in milk and beer until smooth. Simmer 2-3 minutes until slightly thickened.
Remove from heat, stir in shredded cheddar until melted. Add Dijon mustard, garlic powder, smoked paprika, and season with salt and pepper.
Serve warm beer cheese dip with fresh pretzels immediately.
Notes
Use warm beer (not hot) to avoid killing yeast. Brush pretzels with melted butter after baking for extra shine. Beer cheese dip will thicken as it cools; reheat gently with a splash of milk if needed.