In a small bowl, with a fork, blend butter and 1 tablespoon mustard.
Combine wine and shallots in a 1 1/2- to 2-quart pan; stir often over medium-high heat until liquid is almost evaporated, 5 to 6 minutes. Whisk in cream and remaining 1 tablespoon mustard. Simmer, stirring often, until sauce is reduced to 1 cup and is thick enough to coat spoon in a velvety layer, 8 to 10 minutes. Add salt and pepper to taste.
Meanwhile, in a 5- to 6-quart pan over high heat, bring 4 inches of water to a boil. With a slotted spoon, immerse each whole egg in water for 10 seconds. Lift out. Pour out all but 1 inch water; reduce heat so bubbles on pan bottom pop to the surface only occasionally.
Spread 1 1/2 teaspoons mustard butter on one side of each slice of bread. Lay, buttered side down, on a griddle or in two 10- to 12-inch frying pans over medium-high heat; spread remaining butter mixture over tops of slices. When bottoms are lightly browned, in about 3 minutes, turn slices over. Top equally with cheese. Reduce heat to medium.
One at a time, crack eggs and, holding close to surface of water, break open and let egg slide in. Cook to desired doneness (poke with a spoon to check), 4 minutes for soft yolks.
When cheese on sandwiches begins to melt, in 2 to 3 minutes, top with prosciutto; cook until warm, about 2 minutes.
Mound salad mix on plates. Transfer a sandwich to each mound. With a slotted spoon, lift poached eggs from water and set two on each sandwich. Ladle mustard-wine sauce over eggs. Serve immediately, adding more salt and pepper to taste.
Notes: The mustard butter and mustard-wine sauce (steps 1 and 2) can be made up to 1 day ahead; cover each and chill. Warm sauce over low heat or reheat in a microwave oven at full power (100%), stirring occasionally, to use. Garnish sandwiches with fresh chive spears.
Nutritional analysis per sandwich.