In a large skillet, heat 1 tablespoon each of butter and olive oil over medium heat. Add thinly sliced onions with a pinch of salt, and cook for about 20-25 minutes, stirring occasionally until golden brown.

Preheat your oven to 200°C (400°F). Toss sliced sweet potatoes with olive oil, salt, and pepper on a baking sheet. Roast for approximately 15 minutes until tender and slightly browned.

On a floured surface, roll out the chilled puff pastry into a circle, about ¼ inch thick. Transfer to parchment paper and spread caramelized onions evenly, leaving a 2-inch border.

Arrange roasted sweet potato slices in an overlapping pattern over the onions. Crumble feta cheese and sprinkle thyme leaves over the top.

Fold the edges of the pastry over the filling, creating pleats, and brush with a beaten egg for a shiny finish.

Bake in the preheated oven for 25-30 minutes until the pastry is golden brown and puffed up.