Caramelize the onions: In a large skillet, combine butter and olive oil over medium heat. Add onions and a pinch of salt, stirring occasionally for 20-25 minutes until deep golden brown.

Roast the sweet potatoes: Preheat the oven to 200°C (400°F). Slice sweet potatoes, toss in olive oil, salt, and pepper. Spread in a single layer and roast for about 15 minutes.

Prepare the puff pastry: Unroll chilled puff pastry onto parchment paper, shape into a rough circle about 12 inches in diameter, leaving a 2-inch border.

Assemble the filling: Spread caramelized onions on the pastry, layer roasted sweet potatoes on top, crumble feta cheese, and sprinkle fresh thyme.

Fold the edges: Fold the pastry edges over the filling, creating pleats. Brush with beaten egg for a golden finish.

Bake the galette: Transfer to a baking sheet and bake for 25-30 minutes until puffed and golden brown.

Serve warm: Let cool for a few minutes, then cut into wedges and serve warm with fresh thyme or lemon zest.