Unroll puff pastry; press into fluted 9-inch (23 cm) tart pan with removable bottom. Line with foil; fill with pie weights or dried beans. Bake on bottom rack in 400 F (200 C) oven for 20 minutes. Remove weights and foil; let cool in pan on rack.
Meanwhile, in skillet, heat oil over medium-high heat; cook onions, stirring, until golden, about 3 minutes. Add 1/4 cup water; reduce heat to medium. Cover and cook, stirring occasionally, until no liquid remains and onions are soft, about 5 to 8 minutes.
Stir in vinegar, honey and half each of the salt and pepper; cook, stirring, until onions are dark golden, about 8 minutes. Remove from heat; stir in 3/4 tsp of the thyme. Set aside.
Stir together egg, cream, mustard, and remaining salt and pepper. Sprinkle bottom of pastry with half of the Gruyère or Comté cheese; top with onion mixture. Pour egg mixture over top; sprinkle with remaining cheese and thyme.
Bake in 400 F (200 C) oven until crust is golden and knife inserted in centre comes out clean, about 25 minutes. Let stand for 10 minutes before cutting into wedges.