Preheat the broiler on low.
Gently dry the bread under the broiler, just until dry and with no browning.
In a small bowl, toss the cheese with the white wine and allow to sit.
Meanwhile, in a small bowl, mix together the sour cream, garlic, mustard, salt, and pepper. On a baking sheet, lay out the bread slices. Take less than one-quarter of the sour cream mixture and divide evenly among the bread slices, spreading thinly to coat 1 side of each slice. Next, evenly distribute the ham on top of the bread. Divide the rest of the sour cream mixture over the ham. Spoon the cheese on top of the sour cream mixture, blotting lightly before adding to remove any excess wine.
Preheat a large, nonstick saute pan over medium heat and melt the butter. Crack the eggs into the pan and fry, either over-easy or sunny-side up, according to your preference.
While the eggs are cooking, place the open-faced sandwiches under the broiler until very bubbly and lightly browned, about 3 minutes. Watch closely and don't let the topping burn. Place each sandwich on a plate. Season the fried eggs with salt and pepper and place 1 egg on each sandwich. (If desired, use a biscuit cutter to cut the egg into an evenly round shape.)