Preheat the oven to 400 degrees F. Combine the cornichons and brine, radish, shallot, 1 tablespoon vinegar, 1/4 teaspoon salt and a few of grinds of pepper in a medium bowl; set aside. Whisk the remaining 2 tablespoons vinegar and the mustard in a separate bowl. Slowly whisk in the olive oil and season with salt and pepper; set aside.
Wrap a slice of bacon around each hot dog in a spiral. Heat a large skillet over medium heat. Add the hot dogs and cook, giving them a quarter turn occasionally, until the bacon is crisp, 10 to 15 minutes.
Split open the croissants, leaving the two halves attached at the hinge. Place opened up on a baking sheet and top with the cheese. Bake until the cheese is melted, about 5 minutes.
Fill each croissant with a hot dog; top with some frisee and cornichon relish and drizzle generously with the vinaigrette. Serve with more mustard.