Make mornay sauce. Melt butter in a medium sauce pot over medium heat. Add flour, whisking to combine. Slowly add the milk, whisking constantly. Bring to a simmer and cook until thickened, stirring frequently. Remove the sauce from heat and stir in gruyere, mustard, and nutmeg, mixing until cheese has melted. Season with salt and set aside.
Prepare ham katsu. Place flour in a shallow bowl, eggs in another bowl, and panko in a third bowl. Stack two slices of ham and dredge in flour, shaking off the excess. Dip the ham in eggs, and coat in panko, making sure to evenly coat all sides. Repeat with remaining ham.
Add enough oil in a large saute pan to come up 1-inch up the sides. Heat over medium-high heat until the temperature reaches 340 degrees F. Carefully lay the breaded ham in the oil and fry until golden brown. You may have to cook the ham in batches. Remove ham from the oil and place on paper towels to drain excess oil.
Assemble sandwiches. Spread butter on each of the bread slices. Flip half of the slices over so that the buttered side is facing down. Top with ham katsu and grated gruyere. Place the remaining bread on top, with the buttered side facing up.
Heat a large nonstick saute pan over medium heat. Lay the sandwiches and cook until golden brown, about 2 minutes. Flip and cook until golden brown on the other side. Remove from heat.
Preheat broiler to high.
Lay sandwiches on a baking sheet and spread about ¼ cup mornay sauce on top. Broil until the sauce bubbly and has seeped into the bread, about 2 minutes.
Meanwhile, fry the eggs until desired degree of doneness. Top the sandwiches with fried eggs, chopped chives, and a sprinkle of togarashi. Serve immediately.