Heat the oil in a large skillet over medium heat. When the oil is hot, add the mushrooms, garlic and thyme. Cook, stirring occasionally, until the mushrooms have browned, 5 to 6 minutes. Season with a pinch of salt and pepper. Remove the skillet from the heat.
Press the pie crust into a deep, 9-inch tart pan with a removeable bottom (or a deep-dish pie pan). Layer the mushrooms and spinach into the crust and top with the cheese.
Whisk together the half-and-half, eggs and a good pinch of salt and pepper in a large bowl. Pour the egg mixture into the crust.
Place the pan on a rimmed baking sheet. Lightly cover the pan with aluminum foil and bake for 45 minutes. Remove the foil and continue baking until the quiche is set and the crust is golden brown, another 10 to 15 minutes. Let the quiche sit for 10 to 15 minutes, then remove from the pan, slice and serve.
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