Combine the shrimp and kosher salt in a food processor and pulse until the shrimp are coarsely chopped. Add the egg, cream, Worcestershire sauce, hot sauce and Cajun seasoning and process until smooth.
Combine the crawfish, corn and green onions in a large bowl. Fold in the puréed shrimp mixture. Divide the mixture into twelve 1/3-cup portions.
With moistened hands shape the mixture into 1 1/2 x 3-inch logs and arrange on a baking sheet. Insert a popsicle stick into one end of each log, pushing the stick almost to the other end. Cover the corn dogs and chill in the refrigerator for 4 to 6 hours.
Mix the flour, cornmeal, paprika and 1 teaspoon salt in a medium bowl. Whisk in the beer. Dip one corn dog at a time in the batter, coating the corn dog and 1/2-inch of the popsicle stick evenly. Deep-fry the corn dogs 4 at a time in peanut oil heated to 350°F Drain on paper towels and serve immediately.
To Make The Spicy Mustard: Combine the Dijon mustard, mayonnaise, chives, Worcestershire sauce, hot sauce, and dry mustard in a medium bowl and mix well. Season with salt and pepper.