Melt butter in a small pan over medium to medium-high heat until the foam subsides. Sprinkle in flour and whisk; add milk and whisk to thicken sauce. Season with salt, pepper and nutmeg; remove from heat.
Meanwhile, simmer dogs in a half inch of water in a nonstick skillet to heat through, 3-5 minutes. Remove from water, dry then butterfly open. Pour the water out of pan and return it to medium-high heat. Add a drizzle of oil and brown and crisp the butterflied hot dogs on both sides. Cut the hot dogs in half across.
Heat a Panini press, griddle pan or large, cast-iron skillet. (If making panini, skip the next step.) Lightly spread one side of the sliced bread with mayonnaise or soft butter. Build the sandwiches with greased sides of bread facing out.
Layer the bread with béchamel sauce, cheese, a grilled hot dog (set the halved dogs side by side to cover bread in single layer, 1 dog per sandwich), more cheese and some crispy onions. Slather Dijon on 1 side of four slices of bread, set some sliced cornichons into the mustard then set bread tops in place with the Dijon facing in and the mayo/butter facing out, if using.
Grill or press sandwiches until deeply golden and cheese has melted. Cut corner to corner and serve with a simple green salad of upland cress or arugula dressed with white balsamic or wine vinegar and olive oil, flaky salt, and crispy, thin-cut oven frites/fries tossed with minced garlic and herbs like parsley and thyme, and finished with flaky salt.