01 - Preheat oven to 375°F. Roll out pastry to fit a 9-inch tart pan. Press pastry into pan, trim excess edges, and prick base evenly with a fork. Chill in refrigerator for 10 minutes.
02 - Line pastry with parchment and fill with baking beans. Bake for 15 minutes, then remove beans and parchment and bake 5 additional minutes until base is pale golden. Remove from oven and set aside.
03 - Heat olive oil and butter in a large skillet over medium-low heat. Add sliced onions, sugar, salt, and pepper. Cook, stirring frequently, for 20 to 25 minutes until onions are soft and golden. Stir in thyme and cook 2 more minutes. Remove from heat and let cool.
04 - In a bowl, whisk together eggs, heavy cream, whole milk, ground nutmeg, salt, and pepper until smooth.
05 - Evenly spread caramelized onions on the baked pastry base. Scatter crumbled goat cheese over onions. Pour the custard mixture carefully over filling.
06 - Bake in preheated oven for 25 to 30 minutes until custard is set and golden on top. Allow to cool for 10 minutes before serving.
07 - Decorate with extra thyme sprigs and fresh arugula leaves if desired. Serve warm or at room temperature.