Place the wood chips in a bowl and cover with cool water. Set aside to soak for 1 hour.
In the meantime, place the salt, sugar, water, and red pepper flakes in an electric kettle and heat just until the salt and sugar have dissolved. Turn the kettle off and add the ice. Stir until the brine has cooled. Place the shrimp into a 1-gallon zip-top bag and add the brine. Place in a leak-proof container and set in the refrigerator until the wood chips are done soaking, approximately 45 more minutes.
While the shrimp are brining, heat about 2 quarts of charcoal in a chimney starter until the coals are covered with a white ash.
After the chips are done soaking, drain them and set aside. Drain the shrimp in a colander and thread on 4 to 6 rounded skewers.
Carefully transfer the charcoal to an 18-inch round stainless-steel bowl. Cover the charcoal with the wood chips. Lay the skewered shrimp directly atop the chips, making sure that the shrimp do not come in direct contact with the charcoal and that the shrimp are in one layer. Turn another, smaller (13- to 14-inch) metal bowl upside down on top of the shrimp. Cook until the shrimp are firm and no longer translucent, 15 to 20 minutes. Serve warm or chilled with Spicy Brown Mustard Dipping Sauce.