In a small bowl, mix together the sour cream, garlic, mustard, salt, and pepper. Let stand at lets 15 minutes or longer.
Preheat the broiler on low.
Gently dry the bread under the broiler,on one side only just until dry and with no browning.
In a small bowl mix the mayo and butter until completely incorporated. Spread some of the mayo butter on the undried side of the bread.
Heat a griddle or large fry pan on medium low and place bread in pan butter side down to brown. Make sure to get a good crusty brown. Remove to baking sheet.
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.
Reheat the griddle or pan you used to brown the bread in, over medium heat and melt 1 tablespoon of 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. Serve