Lay 1 sheet of phyllo on a work surface (cover the remaining phyllo with a slightly damp kitchen towel to keep it from drying out). Spray the phyllo sheet generously with cooking spray.
Top with another sheet of phyllo. Spray the second sheet of phyllo with cooking spray. Cut the phyllo stack lengthwise into 3 strips (each strip should measure about 3×9 inches). Place 1 piece of hot dog at the end of each strip.
Fold the sides of the phyllo in to overlap the edges of the hot dog pieces. Roll the hot dogs up tightly to encase them in phyllo. Place the rolled hot dogs, seam side down, on the prepared baking sheet.
Repeat the process with the remaining hot dogs and phyllo. When all of the hot- dog packages have been assembled, spray them lightly with cooking spray, and bake until they are golden brown and crispy, about 16 minutes.
Meanwhile, bring the milk to a boil in a small nonstick saucepan over high heat. Add the cheese and whisk until it melts. Continue whisking until the sauce is smooth. Whisk in the ketchup and mustard.
Serve the hot dogs with the warm sauce for dipping.