Remove the stems from the mushrooms and finely chop them.Reserve the caps.
In a medium skillet, melt the butter in the oil over moderately high heat.
Add the chopped mushroom stems and cook, stirring often, until nicely browned, about 5 minutes.
Reduce the heat to moderately low, add the onion, thyme and pepper and cook for 2 minutes longer.
Stir in vermouth and cook until evaporated, about 1 minute.
Scrape into a medium bowl.
With a fork, stir in the tuna, bread crumbs and 1 tablespoon of water. Blend in the Parmesan cheese.
Stuff the mushroom caps firmly with the filling, rounding the tops slightly.
When ready to serve, preheat the oven to 400 degrees.
Place the stuffed mushrooms on baking sheet and bake for about 15 minutes, until hot and golden brown on top.