Heat olive oil over medium-low heat in a large skillet.

Add sliced onions, ½ teaspoon salt, and sugar.

Cook slowly, stirring occasionally, until the onions are deep golden and caramelized, about 20-25 minutes.

Stir in balsamic vinegar and cook for an additional 1-2 minutes until fully absorbed; set aside to cool.

In a separate skillet over medium heat, crumble chicken apple sausage (without casings) and cook until well-browned and cooked through, 5-7 minutes.

Break up with a spatula for even browning.

Remove from heat and let cool slightly.

On a clean work surface, lay out thawed puff pastry.

Evenly distribute the caramelized onions over the center, leaving a 2-inch border around the edges.

Top with cooked sausage, then sprinkle cheddar cheese and thyme leaves over the mixture.

Fold edges of the pastry up and over the filling, pleating as necessary to create rustic folds—leave the center exposed.

Beat the egg with a teaspoon of water, then brush the egg wash over the pastry edges to encourage a glossy, golden finish.

Carefully transfer galette to a parchment-lined baking sheet.

Bake in a preheated 400°F (200°C) oven for 25-30 minutes or until the pastry is deeply golden and crisp.

Allow to cool slightly before slicing.