For the pretzels:

1 1/2 cups warm water
2 tablespoons light brown sugar
1 package active dry yeast (2 1/4 teaspoons)
3 ounces unsalted butter, melted
2 1/2 teaspoons sea salt
4 1/2 to 5 cups all-purpose flour
2 tablespoons chopped fresh rosemary
Canola oil, to grease bowl
3 quarts water
2/3 cups baking soda
1 whole egg, beaten with 1 tablespoon cold water
Coarse sea salt

For the Rosemary Cheese Sauce:

2 tablespoons butter
2 tablespoons all-purpose flour
1 cup milk
1 cup shredded Cheddar cheese
2 teaspoons fresh rosemary
Salt and pepper, to taste