Heat a large saucepan over medium heat. Add the olive oil and carefully add the thyme bundle. Once it crackles and becomes crispy set aside on a paper towel to drain. Add the diced onion, and salt and pepper, to taste, and cook until tender, about 3 to 5 minutes. Add the grated garlic and cumin and lower the heat. Stir in the sliced mushrooms, and season with salt and pepper, to taste. Cook until the mushrooms are tender and most of the liquid has evaporated at the bottom of the pan, 8 to 10 minutes.
Stir the vermouth into the mushrooms and cook until the flavor of the alcohol has mellowed considerably, 5 to 8 minutes. Taste for seasoning. Add the sour cream and fried thyme back to the pan. Reduce heat and allow sour cream to melt over the mushrooms. Do not overheat the cream or it will separate. Stir in half of the grated Parmesan. Taste for seasoning.
Heat a medium saute pan over medium heat and add the butter and the mushroom bottoms. Season with salt and pepper, to taste, and cook over medium heat, until the water emerges from the mushrooms and evaporates. 5 to 8 minutes, Stir in the vermouth and cook 5 to 8 minutes. Taste for seasoning, then set aside to cool.
Finish the dish:
Preheat the oven to 350 degrees F.
When the mushrooms have cooled slightly fill each bottom with some of the stuffing. The stuffing should be firmly packed in the mushroom bottom well above the top. Arrange the mushrooms, in a single layer, on a baking sheet. In a small bowl, mix the bread crumbs and remaining Parmesan, then sprinkle it on top of the mushroom caps. Heat in the oven until they are hot inside and lightly browned on the top, 5 to 8 minutes. Transfer the mushrooms to a serving platter and drizzle each cap with a little balsamic vinegar. Serve immediately.