½ cup mayonnaise
2 tablespoons dijon mustard
4 slices sourdough or country-style bread
1 cup packed finely grated firm swiss cheese, such as Emmental
2 tablespoons unsalted butter
3 ounces thinly sliced prosciutto
4 large eggs
Make the toast soldiers: In a small bowl, whisk together the mayonnaise and mustard. Spread 1 teaspoon of the dijon mayo onto each side of the slices. Using a serrated knife, cut each slice of bread into 1 x 3 inch wide sticks. Place the grated cheese in a shallow dish. Lightly roll the bread sticks in the cheese, and wrap one slice of prosciutto around each stick.
Place a cast iron skillet over medium-high heat , and melt the butter. Arrange the soldiers in a single layer in the skillet. Cook the bread sticks for 2 minutes and flip them over. Cook them on the second side for another 2 minutes, or until the bread is golden and crisp, and place them on a plate covered with a paper towel.
Make the eggs: Fill a large bowl with icy water and set aside. Bring a medium saucepan with 4 inches of cold water to a boil, then turn it down to a rapid simmer. Place 4 eggs in the water, and allow them to cook for 5 ½ minutes. Use a slotted spoon to remove the eggs from the water, and place them in the ice bath for 30-60 seconds. Place the eggs in dipping egg cups. Tap each shell lightly with a small spoon and remove the tops of the shells. Season the eggs with sea salt and freshly cracked black pepper. Serve with croque monsieur toast soldiers for dipping.