01 - Preheat oven to 400°F. Line a 9-inch tart pan with parchment paper. Roll puff pastry and press into the tart pan. Trim excess and prick base with fork. Chill while preparing filling.
02 - Heat olive oil and butter over medium heat in a skillet. Add onions and cook, stirring occasionally, until soft and golden, about 12 minutes.
03 - Add mushrooms to onions, cooking 8-10 minutes until moisture evaporates and mushrooms brown. Stir in garlic and thyme, cook 1 more minute. Remove from heat and cool slightly.
04 - Whisk eggs, heavy cream, salt, and pepper in a bowl. Fold in grated cheese until combined.
05 - Spread onion-mushroom mixture evenly over chilled pastry shell. Pour egg and cheese mixture on top.
06 - Bake tart for 25-30 minutes until filling is set and pastry is golden brown.
07 - Let tart cool for 10 minutes. Garnish with chopped fresh chives or parsley. Slice and serve warm or at room temperature.