Preheat oven to 400. Combine butter and crushed garlic. Set Aside. Line a cookie sheet with parchment.
Put cheese in a microwave-safe bowl. Microwave one minute. Stir. Microwave 30 seconds. Stir. At this point, all the cheese should be melted. Microwave 30 more seconds until uniform and gloopy (it should look like cheese fondue at this point). Mix in the egg and dry ingredients. You will need to dump it onto wax paper and knead it by hand to thoroughly incorporate the ingredients or you can do this in a food processor with the dough blade.
Press into a big rectangle on parchment paper (it helps to wet your hands). Cut into 8 long strips. Cut each strip into 3 smaller strips. You will end up with 24 strips of dough. Attach the tops of 3 strips. Braid the dough. Attach the bottoms and tuck under. Put the braid on a parchment-lined baking sheet. Continue to make 8 braided breadsticks.
Brush each with the garlic butter (using ½ of it now). You want to get some of the little pieces of garlic on them too. Sprinkle a little sea salt on the breadsticks.
Bake for 15 minutes. Remove from the oven and brush with the rest of the garlic butter. Bake for another 5-10 minutes or until golden brown. Serve warm.