Preheat oven to 180°C.
Toast bread cubes in in a large shallow baking pan until golden-brown, about 10 minutes.
Heat the butter in a medium-large frying pan. Add the chopped onion and fry over a medium heat, stirring occasionally, until it begins to soften.
Add the chopped mushrooms, minced garlic, 1/2 teaspoon salt and 1/4 teaspoon freshly milled pepper. Continue to fry and stir until the liquid the mushrooms give off has evaporated (about 15 minutes).
Add the white wine and parsley and continue to fry and stir for another few minutes.
Check the seasoning and remove from the heat.
In a medium-large bowl, combine the milk, creme fraiche, single cream, eggs and grated Parmesan cheese, 1/2 teaspoon salt and 1/4 teaspoon freshly milled pepper.
Whisk together well.
Stir in the toasted bread cubes and the mushroom mixture until coated well and let stand for 10 minutes to allow the bread to absorb some of the cream and egg mixture.
Butter the baking dish. I used a square dish (20 x 20 cm), about 8 cm deep.
Spoon the mixture into the baking dish and place on a rack in the lower third of the oven. Bake for 30 to 35 minutes until nicely browned on top. Please note that the baking time will depend on the depth of the baking dish you have chosen.