Cook potatoes in boiling salted water until tender and drain. Return to pot and cover to keep warm while preparing mushrooms.
In a sauté pan over medium high heat, melt butter and heat oil. Add mushrooms and shallots and sauté until moisture has evaporated, about 5 minutes.
Add garlic and thyme and sauté one minute more. Add vermouth and cook until absorbed. Remove from heat.
Roughly mash potato and sour cream with a fork. Stir in mushroom mixture and Asiago and season to taste. Set aside until ready to serve.
To serve, preheat oven to 400°F. Spoon potato mixture onto toasted baguette slices. Heat for about 7 minutes, until warmed through. Serve warm or at room temperature.