Preheat oven to 400 degrees F.
Using a mandolin or super-sharp knife, cut onion into paper thin slices. Clean and slice the mushrooms.
Heat 1 tablespoon oil in a skillet over medium heat; add onion and saute until caramelized. Transfer to a bowl and set aside. Heat remaining teaspoon of oil in skillet over medium-high heat. Add mushrooms and saute until they release all their liquid and most of it boils away, should take about 5 minutes. Combine caramelized onions with mushrooms and saute together briefly; season with salt and pepper. Remove the pan from heat.
Unfold puff pastry onto a floured surface; cut in half lengthwise to form 2 long rectangles. Gently roll out each rectangle to approximately 5 by 14 inches and place on cookie sheet. Trim edges by 1/4-inch strips all around; set strips aside. Break 1 egg into a small bowl; beat slightly. Brush edges of pastry with some of the egg. Use trimmed strips to make a raised border on each. Brush entire surface with remaining beaten egg. Prick the interior of pastry all over with a fork. Bake until pale gold, about 10 minutes. If pastry has puffed up inside edge, press it down gently. Set aside.
Meanwhile, combine remaining eggs with 6 ounces goat cheese and blend until smooth. Spread the mixture onto pastry. Return to oven and bake just until set, about 4 minutes. Add the thyme to the mushroom mixture. Remove cookie sheet from oven and spread with thyme-mushroom mixture. Crumble remaining cheese on top. Just before serving, broil tarts on low heat for 2 to 3 minutes, until tarts are warm and cheese softens.
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.
Recipe courtesy of David Nash