Heat oven to 400°F. Line a baking sheet with parchment.
Warm olive oil and butter in a large skillet over medium heat.
Add onions and 1/2 teaspoon salt. Cook, stirring occasionally, until deeply golden, 25–30 minutes.
Stir in brown sugar and balsamic. Cook 2 minutes to glaze. Add pepper and half the thyme.
Lightly flour the counter. Roll puff pastry to smooth seams and slightly enlarge.
Cut pastry into eight squares. Transfer to the sheet and dock centers with a fork.
Whisk egg with water. Brush pastry borders, avoiding the centers.
Crumble goat cheese onto centers. Spoon a mound of onions over the cheese.
Bake until puffed and deeply golden, 18–22 minutes.
Rest 5 minutes. Sprinkle remaining thyme and drizzle honey or extra balsamic. Serve warm.