To make the mushroom pudding, preheat the oven to 350 degrees.
Using 1 tablespoon of unsalted butter, lightly butter an 8 x 8 x 2-inch glass baking dish.
To prepare the mushrooms, remove the stems from the shiitakes and slice each mushroom into thirds.
In a large saute pan over medium-high heat, melt the butter and oil together.
Add the mushrooms in batches and cook until browned.
Return all the mushrooms to the saute pan and add the garlic, ginger, lemongrass and basil and cook one minute, stirring to blend.
Remove mushrooms from the pan to a bowl.
Add salt and pepper to taste.
If there is any liquid left from the mushrooms, remove to a small cup and set aside.
In a large bowl with a whisk, place the half-and-half, eggs and any leftover liquid from the mushrooms and whisk until blended.
Add the cheeses and salt and pepper.
Add bread cubes and let stand for 15-20 minutes for bread to soak up the liquid.
Stir in mushroom mixture and pour into baking dish.
Bake until lightly browned and puffy, approximately one hour.
Remove from oven and sprinkle with 1 tablespoon of Parmesan cheese and the chopped chives.