In a small saucepan, melt 2 tablespoons butter over medium heat. Whisk in the flour and cook for 1 minute. Whisk in the milk until thickened, about 5 minutes. Season the sauce with salt to taste. Reserve one-quarter of the sauce for the dogs sammy. Season the remaining sauce with the pepper and nutmeg and whisk in the mustard.
Spread one side each of 6 bread slices with the peoples sauce. Use to make 3 sammies with 2 slices each of ham and cheese. Spread the remaining 2 bread slices with the dogs sauce and then assemble the sammy using the remaining ham and cheese; keep separate.
In a large skillet, melt the remaining 2 tablespoons butter over medium-low heat. Add the sammies and cook, turning once, until golden on both sides, about 4 minutes total. Chop up the pups sammy and let cool, but eat yours hot!