Cook mushrooms and shallot in butter with salt and pepper in a large, heavy
skillet over moderately high heat, stirring, until mushrooms are tender and
the liquid they give off is evaporated, about 10 minutes. Remove from heat and stir
in crème fraiche and tarragon.
Fit 1 slice of ham into 12 lightly oiled muffin cups (ends will stick up and
hang over edges of cups). Divide mushrooms among cups and crack 1 egg into each.
Bake in middle of oven until whites are cooked but yolks are still runny, about
15 minutes. Season eggs with salt and pepper and remove (with ham) from muffin
cups carefully, using 2 spoons or small spatulas.