For the oil: Using your palm or the back of a chef knife, gently crush the garlic until it is slightly split, but still intact. Place the garlic into an airtight container and cover with olive oil. Seal the container with a lid and let it steep overnight prior to using.
For the dough: In a large bowl, mix together the flour and salt. With your hands, cut the butter into the flour until crumbles, the size of peas, appear. Mix in 3 tablespoons cold water, 1 tablespoon at a time, until a dough forms. Dump on a board and knead into a ball. Wrap the dough in plastic and rest it in the refrigerator for 30 minutes.
For the filling: Preheat the oven to 400 degrees F. Brush a baking tray with some garlic oil and sprinkle with salt. Spread the mushrooms, shallots and thyme on the baking sheet and roast until the mushrooms are tender and browned, 15 to 18 minutes. Remove from the oven and pour 2 tablespoons of canned mandarin orange juice over the mushrooms. Using a flat spatula, loosen the mushrooms, shallots and any brown bits stuck on tray. Set aside to cool.
On a lightly floured surface, roll the dough into a 10-inch circle. Lay the dough onto a baking tray. Toss the mushroom mixture with the cheese in a large bowl and spread in the center of the dough. Fold the dough over the filling, leaving a 4-inch opening in the center. Arrange the mandarin orange segments in the center opening. Brush the dough with the beaten egg. Bake until the dough is golden brown and the filling is bubbling, 35 to 40 minutes. Slice and serve warm.
Trumpet oyster mushrooms can be found at your local farmer's market. The garlic oil will keep for at least 1 month at room temperature.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.