Soft Pretzels:
1½ cups warm water
2 tablespoons light brown sugar
1 package active dry yeast
3 ounces unsalted butter, melted
2½ teaspoons kosher salt
4½ to 5 cups all-purpose flour
Vegetable oil
3 quarts water
¾ cup baking soda
1 whole egg, beaten with 1 tablespoon cold water
Coarse sea salt

For the cheese sauce:
½ Tablespoon unsalted butter
½ Tablespoon all-purpose flour
½ cup milk
8 ounces Cheddar cheese, grated