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Preheat oven to 400° F (200°C).
Heat oil in a skillet and melt butter. Add onion and sauté until soft. Add mushrooms and cook until all liquid has evaporated. Season to taste and remove from heat. Stir in tarragon and cream.
Lightly oil 12 muffin cups and line cups with ham slices, leaving ends to stick up and hang over edges. Divide mushroom mixture among muffin cups and crack an egg into each one.
Sprinkle with salt and pepper. Bake in oven until egg whites are cooked and yolks are still soft, approximately 12 – 15 minutes. Remove from oven and loosen edges with a knife or spatula. Transfer 2 ham cups to each plate. Serve with brioche or warm buttered toasts.
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