Put oven rack in middle position and preheat oven to 375°F (or as per pie shell directions). Bake pie shell following package directions.
Cool completely in pan on a rack, about 15 minutes.
Make filling while shell bakes:
Heat butter and oil in a 12-inch heavy skillet over moderately high heat until foam subsides, then sauté mushrooms, shallot, thyme, 1/2 teaspoon salt, and 1/4 teaspoon pepper, stirring frequently, until mushrooms are tender and any liquid given off is evaporated, 8 to 10 minutes. Transfer to a bowl and cool to room temperature.
Whisk together crème fraîche, heavy cream, whole egg, yolk, and remaining 1/4 teaspoon salt and 1/8 teaspoon pepper in a medium bowl until combined.
Fill and bake tart:
Reduce oven temperature to 325°F
Scatter mushrooms evenly in tart shell and pour custard over them. Bake tart in pan on a baking sheet until custard is just set and slightly puffed, 35 to 45 minutes.
Cool tart in pan on rack at least 20 minutes, then remove side of pan. Serve tart warm or at room temperature.