Thaw frozen puff pastry 3-4 hours at room temperature. (Must still be cold)
Pre-heat oven to 400° F (200°C). Line a baking sheet with parchment paper.
Prepare mixture. Melt butter and olive oil in a heavy skillet over medium high heat. Add the sliced onions all at once, stirring to coat the onions. Allow onions to cook, undisturbed, for about 4 minutes. Add salt, thyme, pepper, and sugar, and stir. Reduce heat to low; cover and cook, stirring occasionally, until the onions are deep brown and very soft, about 15 min. Stir in broth (or beer or wine), scraping any brown bits from the bottom of the pan. Let cool completely.
Preparing pastry. Roll out the thawed Tenderflake® Puff Pastry on a lightly floured work surface. Use a 2½ “inch (7cm) round cookie cutter to cut circles out of the pastry. Two sheets should produce 24 pastry rounds.
Brush each round lightly with egg. Top 12 of the pastry rounds with a scant tablespoonful of caramelized onions and 1 teaspoonful grated cheese.
Carefully place the remaining pastry rounds over the onion and cheese mixture. Use the tines of a fork around the edge of each pastry round to secure the pastry together.
Arrange the filled pastries on the prepared sheet about 2” inches (5 cm) apart and brush lightly with egg. Prick the top of each pastry with the tines of a fork for ventilation and sprinkle with coarse salt.
Bake for 12 to 15 minutes, or until puffed and dark golden brown. Remove from the oven and allow to cool for at least 5 minutes before serving.