Preheat the oven to 350 degrees F. Spread 4 bread slices with the mustard and the other 4 with the preserves. Toss both cheeses in a bowl, then divide half the mixture among the mustard-covered bread. Top with the ham, then the remaining cheese and the preserves-covered bread. Press each sandwich slightly. Whisk the eggs, milk, 1/2 teaspoon salt and a few grinds of pepper in a baking dish.
Melt 1 1/2 tablespoons butter in a large nonstick skillet over medium-high heat until just starting to brown. Dip 2 sandwiches in the egg mixture, turning to coat. Let the excess drip off, then transfer to the hot skillet and cook until browned and crisp, about 2 minutes per side. Transfer the sandwiches to a rack set on a baking sheet. Wipe out the skillet, add the remaining 1 1/2 tablespoons butter and cook the remaining 2 sandwiches. Transfer to the rack, then transfer the baking sheet to the oven to melt the cheese completely, about 5 minutes.
Meanwhile, toss the greens with the vinegar, a pinch of salt and a few grinds of pepper. Serve with the sandwiches.