Clean mushrooms by wiping with a damp paper towel. Slice mushrooms about 1/4" thick. Peel the onion and slice into thin half-moons. Mince garlic and strip thyme leaves from stems.

Heat a large skillet over medium-high heat. Add olive oil and 1 tablespoon butter. When butter foams, add sliced onions and a pinch of salt; cook, stirring occasionally, until onions are soft and starting to caramelize, about 8–10 minutes.

Push onions to the side, add remaining 1 tablespoon butter. Add mushrooms in a single layer (work in two batches if necessary) and let them brown undisturbed for 2–3 minutes. Stir and continue to cook until mushrooms release moisture and it evaporates, about 5 more minutes.

Add minced garlic and thyme, cook 30–60 seconds until fragrant. Sprinkle the teaspoon of flour over the mixture and stir to combine (this helps thicken the filling). Pour in the 2 tablespoons stock and scrape up any browned bits from the pan. Cook until the mixture is just moist but not saucy, about 1–2 minutes. Season with 1 teaspoon salt and 1/2 teaspoon black pepper. Transfer to a bowl and let cool for 10 minutes.

Cheese & Binding

While filling cools slightly, beat the 4 oz softened cream cheese in a medium bowl until smooth. Stir in 1 cup grated Gruyère (or cheddar), chopped parsley, and the beaten egg. Fold the cooled mushroom-onion mixture into the cheese mixture until evenly combined. Taste and adjust seasoning if necessary.

Pastry & Finish

Position a rack in the center of the oven and preheat to 400°F (200°C). Line a baking sheet with parchment paper.

Unfold or roll each puff pastry sheet on a lightly floured surface to smooth creases. For two-sheet method: cut each sheet into 6 equal rectangles (approx. 3" x 5" each) to make 12 pastries total. Alternatively, cut into 8 rectangles for larger pastries. Keep cut pieces chilled if they become warm.

Place about 2–3 tablespoons of the mushroom-cheese filling in the center of each pastry rectangle, leaving a 1/2" border. Lightly brush the border with a little beaten egg (from the egg wash) to help seal. Fold the pastry over (for turnovers) or fold corners to make a parcel, pressing edges to seal. Use a fork to crimp edges if desired. For open-faced tarts: spoon filling onto the center leaving a 1/2" border and fold the border inward slightly to create a raised edge.

Transfer pastries to the prepared baking sheet, spacing about 1" apart. Whisk the remaining whole egg with 1 teaspoon milk or water and brush the tops and edges of each pastry. If desired, sprinkle a tiny pinch of coarse or flaky sea salt on each pastry for finish.

Cut a small slit in the top of enclosed pastries to vent. Bake in the preheated oven until pastry is puffed and golden, 18–22 minutes for smaller pastries or 22–28 minutes for larger ones. Rotate the sheet halfway through baking for even color.

Remove from oven and let cool on the sheet for 5 minutes, then transfer to a wire rack. Serve warm or at room temperature. These are best the day they are made but can be reheated at 350°F (175°C) for 6–8 minutes to refresh.