Preheat your oven to 400°F and line a baking sheet with parchment paper.
In a large skillet over medium-low heat, melt the butter with olive oil. Add the sliced onions and cook, stirring occasionally, for about 30-35 minutes until they're golden brown and caramelized. (Don't rush this step, mama – it's where all the magic happens!)
Stir in the balsamic vinegar, sugar, thyme, salt, and pepper, cooking for another 2-3 minutes. Remove from heat and let cool slightly.
Roll out the puff pastry on your prepared baking sheet and score a 1-inch border around the edges with a knife (don't cut all the way through).
Brush the border with the beaten egg to get that gorgeous golden finish.
Spread the caramelized onions evenly over the pastry, staying within the border you created.
Sprinkle both cheeses over the onions, distributing them evenly.
Bake for 20-25 minutes until the pastry is puffed and golden brown around the edges.
Let the tart cool for 5 minutes, then garnish with fresh thyme leaves and slice into squares.
Serve warm with a simple arugula salad on the side for the perfect fall dinner!