Mix water and sugar together in a bowl until sugar is dissolved; add yeast. Let mixture stand until the 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 and arrange on a baking sheet; 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.