Preheat oven to 400°F (200°C). Line a baking sheet with parchment.
Sauté shallot in oil and butter until soft, 2 minutes. Add mushrooms, thyme, salt, and pepper; cook until moisture evaporates, about 6–8 minutes. Cool slightly.
Unfold puff pastry and cut into four 4-inch squares. Spread 1/2 tsp Dijon in center of each, leaving a 1/2-inch border.
Divide mushroom mixture among squares, sprinkling each with cheese. Fold edges up and pinch to form rustic rounds.
Brush edges with beaten egg. Bake 15–18 minutes until golden and puffed. Serve warm.