Cut cheddar into thin slices. Butter each slice of bread and spread Dijon mustard on 4 of the slices.
Lay cheddar cheese on 4 buttered slices of bread, covering the whole surface. (You want lots of cheese!) Place bacon or ham on top. Place a slice of bread, Dijon side down, on top of each sandwich.
In a small bowl, whisk eggs, milk or cream together until thoroughly blended and lemony in colour. Whisk in salt and pepper.
Pour half the egg mixture into a shallow plate. Dip each side of two sandwiches in the egg mixture, making sure the bread is saturated. Place on a tray or baking sheet. Repeat with the remaining egg mixture and remaining two sandwiches.
Melt half the butter in a 10-inch frying pan over medium heat. (If you’re cooking on a campfire, set the grill about 8 inches above the flame.) Once the butter is sizzling, place two sandwiches in the pan, cheese side down. Cook for 3 to 4 minutes, turning the pan now and then to brown the bread evenly. Flip and cook for another 3 to 4 minutes, until cheese begins to ooze out the sides.
Remove sandwiches from the pan and tent with tinfoil, or cover with a plate or pot lid to keep warm.
Melt the remaining butter in the pan and cook the remaining two sandwiches the same way, reducing the heat or moving the pan to one side of the flames if it’s getting too hot.
Remove from heat, cut in half and serve at once, with Tomato Tarragon Salsa, ketchup, chili crisp or any other of your favourite condiments. Or go the traditional route: a sprinkle of icing sugar and a dollop of strawberry jam.