In a large skillet over medium heat, heat oil and butter. Add onions; season with salt and pepper and cook, stirring occasionally, until onions are very caramelized, about 35 minutes. Stir in vinegar and thyme. Let cool to room temperature.
Place racks in upper and lower thirds of oven; preheat to 375°. On a lightly floured surface, cut each puff pastry sheet into 9 (3" x 3") squares. Transfer squares to 2 parchment-lined baking sheets. Top each with about 1 tbsp. caramelized onions, leaving a ½" border on all sides. Brush exposed edges with egg. Divide goat cheese among squares.
Bake tarts, rotating sheets front to back halfway through, until puffed and golden brown, 25 to 28 minutes.
Drizzle a little honey on each tart. Top with more thyme and pepper. Serve immediately or at room temperature.
