Step 1

Preheat oven to 350°F. Combine mushrooms, shallot, and garlic in a food processor; pulse until coarsely chopped, about 15 times.

Step 2

Heat oil in a large skillet over medium-high. Add mushroom mixture, salt, and pepper. Cook, stirring occasionally, until mushrooms are golden and fragrant, about 5 minutes. Add sherry and chopped thyme. Cook, stirring constantly, until most sherry evaporates, about 30 seconds. Add cream; bring to a boil. Boil, stirring constantly, until thickened slightly, about 2 minutes. Remove from heat, and add ½ cup of the cheese, stirring until melted.

Step 3

Place phyllo shells on a rimmed baking sheet. Spoon 1 heaping teaspoon of mushroom mixture into each cup, and sprinkle remaining ¼ cup of cheese over shells (about ½ teaspoon per shell). Bake in preheated oven until tops are golden and cheese is melted, about 5 minutes. Sprinkle with chives and additional thyme. Serve warm or at room temperature.