Heat the butter and olive oil in a large skillet over medium heat. Add the sliced onion and about 1/4 teaspoon of the salt, then cook slowly for 15–20 minutes until soft and golden.

Lower the heat if the onion begins to scorch, stirring frequently so it develops a sweet, even color. Add the mushrooms, remaining salt, pepper, and thyme, then cook for 8–12 minutes until the mushrooms release their moisture and the pan is nearly dry.

Stir in the sherry, white wine, or water if using, and cook for another 1–2 minutes. Remove the skillet from the heat and let the mixture cool slightly.

Heat the oven to 400°F (200°C) and lightly grease a mini-muffin pan. Thaw the puff pastry according to the package directions while keeping it cool and workable.

Lightly dust the work surface with flour and roll the pastry just enough to smooth it. Cut the sheet into approximately 24 squares, about 2 1/2 inches each.

Press each pastry square into a mini-muffin well, taking care not to stretch the dough too thin. Add a small amount of goat cheese or cream cheese to each pastry cup.

Spoon the cooled mushroom-onion mixture over the soft cheese, dividing it evenly among the cups. Finish each bite with a little shredded Gruyère, Swiss, or sharp white Cheddar.

Brush the exposed pastry edges with beaten egg for a glossy, deeply golden finish. Keep the assembled pan cool until it is ready to go into the oven.

Bake for 15–20 minutes, or until the pastry is puffed and deeply golden and the cheese is melted. Watch closely during the final minutes because small pastry cups can brown quickly.

Cool the bites in the pan for 5 minutes, then transfer them to a rack. Serve warm or at room temperature.