Preheat the oven to 425 degrees F (220 degrees C). Lightly grease baking trays.

Mix water and sugar together in a bowl until sugar is dissolved; add yeast. Let mixture stand until yeast softens and begins to form a creamy foam, about 5 minutes.

Whisk flour, butter, and salt together in a bowl until well mixed; add yeast mixture and stir with a fork until dough starts to cling. Turn dough onto a floured work surface and knead until smooth.

Cut dough into 15 equal pieces and roll each piece between your hands into rope shapes at least the thickness of a pencil. Twist dough into desired shapes.

Pour 4 cups hot water into a bowl; stir in baking soda until dissolved. Dip each pretzel into the baking soda solution and arrange on baking sheets. Brush with egg and sprinkle coarse salt over each.

Bake in the preheated oven until pretzels are browned and cooked through, 8 to 12 minutes.