In a medium glass bowl, combine mushrooms and Marsala wine. Season with salt and black pepper, to taste. Toss to coat. Marinate for 20 minutes at room temperature. Drain excess moisture before cooking.
While the mushrooms are marinating, slice off both ends of each onion and remove the papery outer layers. Cut in half lengthwise (from tip to root) and then slice each half lengthwise into strips. Tip: Don’t slice too thin or they may burn while cooking.
Melt butter in a large, high-sided skillet set over medium heat. (For best results,do not use a non-stick pan). Add onions and stir until coated with butter.
Cook for approximately 45- 50 minutes, stirring every 5 minutes or so, to ensure they cook evenly and don’t stick to the pan. The exact cooking time will vary depending on several factors, including the age of the onions and how many you cook at one time.
Once the onions are started, heat 2 tablespoons olive oil in another large skillet over medium-high heat. Once the oil is hot, add marinated mushrooms and sauté until browned and tender, approximately 5-7 minutes. Remove from heat and transfer toa bowl. Set aside.
Preheat oven to 375°F and line two large, rimmed baking sheets with parchment paper or Silpat™ baking mats. Set aside.
When the onions reach the desired depth of color, add white wine to deglaze the bottom of the pan. Add more liquid, if necessary, to scrape up all the brown bits and then salt and pepper the onions, to taste. Remove from heat and set aside.
Cut thawed puff pastry dough in half and roll out on a lightly floured surface. With a sharp knife or pizza cutter, cut the puff pastry sheet into 32 equally sized rectangles. (See picture). Transfer puff pastry pieces to a prepared baking sheet and place in the freezer for 10 minutes.
Repeat the above steps for the other half of the puff pastry.
Remove baking sheets from the freezer and let sit for a couple of minutes before brushing each piece with the beaten egg. Divide the grated cheddar cheese, caramelized onions, and sautéed mushrooms among the puff pastry pieces and top with the thyme.
Place baking sheets in the pre-heated oven and bake for 10-15 minutes before rotating each baking sheet to ensure the pastry bites cook evenly.
Continue cooking until the pastry is golden brown, approximately 8-10 minutes. Remove from oven and cool slightly before serving. Enjoy!