Choose your size/shape of pan. Adjust the ingredients accordingly.
Rinse the mushrooms and pat dry. Coarsely chop the mushrooms, garlic, shallots and tarragon.
Lightly coat tarte pan with non-stick cooking spray. Unfold puff pastry and roll out to size of pan allowing for the pastry to go up the sides. Lay in tart pan making sure to press into all corners. Trim excess above top of pan. Perforate (dock) bottom of pastry with fork. Line with parchment paper and fill with pie weights (can also use dried beans or uncooked rice). Bake for 12-15 minutes, or as directed on package, until edges are light golden brown. Remove from oven and remove weights. Let cool on a rack in the tarte pan.
While pastry is baking, heat skillet with butter and olive oil. Saute garlic and shallots until softened being careful to not brown or burn. Add mushrooms and saute on medium low heat, if they soak up the oil and butter add more butter to keep pan just moist. Continue to saute until softened. Remove from heat and add the Sweet Vermouth (can substitute sherry). Return to heat and allow alcohol to burn off. Set aside to cool.
In large bowl beat eggs, add cream and chopped tarragon (reserve 1 tsp of tarragon of chopped or as whole leaves). When mushrooms are fully cooled, fold into egg mixture in small batches. Add salt and pepper to taste.
Pour entire mixture into shell, sprinkle 1/2 of the reserved tarragon on top. Place in oven and cook for 30 minutes or until set. If edges of pastry begin to brown before mixture is set, cover with aluminum foil and continue cooking.
Once cooled slightly, tarte may be removed from pan, cut and served. Sprinkle the remaining tarragon on top as a garnish. Can be served warm or at room temperature but is better slightly warm.
This entire recipe can be done in advance then refrigerated or frozen then reheated in oven to serve.