In a small bowl, combine warm water, yeast, and sugar. Stir gently and let stand until foamy and doubled in size, about 8-10 minutes. If mixture doesn't foam, yeast is dead - start over with fresh yeast.

In a large bowl, whisk together bread flour and salt. Create a well in the center and pour in the activated yeast mixture and melted butter. Mix with a wooden spoon until a shaggy dough forms.

Turn dough onto a lightly floured surface and knead vigorously for 8-10 minutes until smooth, elastic, and slightly tacky but not sticky. The dough should spring back when poked. Form into a ball.

Place dough in a greased bowl, turning to coat all sides. Cover with damp towel and let rise in a warm, draft-free spot until doubled in size, about 60-70 minutes.

Punch down dough and divide into 2 equal portions. Working with one piece at a time, roll into a 24-inch rope, thicker in the middle and tapered at the ends. This requires patience and gentle, consistent pressure.

Shape into pretzel: Form a U-shape, cross the ends twice to create a twist, then fold down and press ends onto the bottom curve of the U. The shaping is crucial for authentic appearance.

Place shaped pretzels on parchment paper and freeze for 30 minutes. This helps them hold their shape during boiling.

Preheat oven to 425°F. In a large pot, bring 5 cups water and baking soda to a rolling boil. Carefully lower one pretzel into boiling solution using a slotted spoon. Boil for 30 seconds, flipping once.

Remove pretzel with slotted spoon, allowing excess water to drain. Place on parchment-lined baking sheet. Repeat with second pretzel. Brush with beaten egg and sprinkle generously with coarse salt.