In a medium saucepan, heat the olive oil until hot but not smoking. Add shallots, and cook over medium heat until translucent, about 3 minutes. Stir in mushrooms, and cook until they are soft and all the liquid is absorbed, about 7 minutes. Stir in salt and pepper. Remove from heat, and set aside to cool. When cool, fold in the creme fraiche and chervil until combined. Set aside.
Preheat oven to 350 degrees. Line a baking sheet with parchment paper. Lay one sheet of phyllo on a clean work surface. Brush phyllo lightly with melted butter, and top immediately with a second sheet of phyllo. Brush again with butter, and top with a third sheet of phyllo. Repeat with remaining 3 sheets of phyllo. Using a pizza cutter, cut the stack of layered phyllo sheets into 24 four-inch squares.
Place a teaspoon of mushroom filling in the center of each square. Carefully draw up the sides of the squares and pinch them together to seal in the filling. Place purses on the parchment-lined baking sheet, and bake until the edges are golden brown, about 20 minutes. Let cool slightly. Tie a chive around each pinched phyllo purse, and top with 1/4 teaspoon caviar. Serve phyllo purses warm.
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