Fry the ham in oil over high heat for 1-2 minutes until crisp, then set aside.
Melt one-third of the butter over medium heat and toast the brioche slices until golden-brown.
Melt another third of butter, add garlic, chestnut, and shiitake mushrooms, and fry for 2-3 minutes.
Lower heat, cover with a lid for 2 minutes, then uncover and fry until liquid evaporates.
Add remaining butter, oyster, and enoki mushrooms, frying for 1-2 minutes until golden.
Season with salt and pepper, then stir in the crème fraîche.
Spoon mushrooms over brioche slices, top with ham and parsley, and serve immediately.