Heat 4 Tbsp. butter in a small or medium Stainless Clad Saucier over medium-high. Once melted, add flour and cook, whisking, until light golden in color, about 2 minutes.

Slowly stream in milk, whisking constantly, until smooth (this is bechamel sauce, one of the French "mother sauces"). Allow to come to a very gentle boil to activate the flour, adjusting heat if necessary. Remove from heat and add 1 cup cheese in a few additions, whisking after each addition, until very smooth. If it looks lumpy at first, just keep whisking (now it’s mornay sauce!) Season with paprika, nutmeg, salt, and pepper. Taste and adjust seasoning. Set aside in a warm place.

Place 8 pieces of bread on a work surface. Spread 4 slices with about a teaspoon or so of Dijon. Top Dijon-slathered slices with ham, pickles, and about half of the mornay sauce and half the cheese. (You are reserving the remaining sauce and cheese for the tops of the sandwiches). Close sandwiches with the remaining slices of bread.

Heat a Carbon Steel Griddle over medium. Preheat broiler to high.

Heat 1 Tbsp. oil and remaining 2 Tbsp. butter on the griddle, smooth with a spatula to combine in a thin layer. Toast sandwiches on both sides until golden brown and cheese is melty, 2–4 minutes per side. Transfer to a Sheet Pan. Top sandwiches with remaining mornay sauce and gruyere.

Add remaining 1 Tbsp. oil to Griddle. Fry eggs to desired doneness. Meanwhile, broil sandwiches watching very closely, until tops are browned and bubbling, 3–5 minutes.

Top each sandwich with a fried egg, a dash of paprika, and chives, if desired. Serve with a green salad on the side.